Stephen Foster was a self-taught instrumentals who received little or no professional training. His bio contains no reference to having been a musician of note.
Billy Joel is a fairly well known pop/rock composer who now writes classical music. Below is a link of him explaining that he can barely play his own compositions and has Richard Joo (of Igudesman & Joo fame) play it in his stead. In other clips I have heard him say that his own brother is a much better piano player than himself- and still he prefers to conduct because he (the brother) does not consider himself a good enough player. (Billy disagreed as I recall.)
In a different context, I was told by some ancient music scholar that Liszt was once visiting Grieg and sight read some unfinished (but apparently complex) piece and judged it to be something to the effect of an amusing little jingle. His point was that Liszt was an amazing pianist, but a bit of a pompous ass. “Something you can get away if you are among the greatest performers of all time” he told me- and he can play a thirteenth (and one or two notes between if they fall where his fingers happen to already be). Apparently Liszt had huge hands and fingers that were very dexterous, but he developed his physical traits to the point he could play non complimentary times in each hand simultaneously. Sort of the opposite of the OP I realize, but an interesting factoid I felt compelled to include.
Link:
My piano teacher, a concert pianist herself, once told me that classical composers weren’t necessarily the best interpreters of their own compositions.
Similarly, Richard Strauss wrote some insanely difficult French horn parts, because he would try them out on his father who happened to be the best horn player in the world.
Andrew Lloyd Webber is a mediocre pianist at best, but he has made a lot of money from his works.
And some guy called Bob Dylan is extremely lucky he started in the 1960’s, because no other decade would have let him sing on stage. Can’t fault his musical composition. though.
Frank Zappa.
And yet:
**Mozart evidently needed a keyboard to work out his musical thoughts. This can be deduced from his letters and other biographical material. For instance, on 1 August 1781, Mozart wrote to his father Leopold concerning his living arrangements in Vienna, where he had recently moved:
"My room that I'm moving to is being prepared—I'm just off now to hire a keyboard, because I can't live there until that's been delivered, especially as I've got to write just now, and there isn't a minute to be lost."[5]
Konrad cites a similar letter written from Paris that indicates that Mozart didn’t compose where he was staying, but visited another home to borrow the keyboard instrument there. Similar evidence is found in early biographies based on Constanze Mozart’s memories.**
Thus, the ability to compose without a keyboard maybe a more modern token of compositional “badassery” than it was in the past. I saw a video a long time ago, maybe something on TV and I think from the 1970s, with a professor or composer sneering at composers who could not write without an instrument, saying he could tell the difference. Oh well, I guess Mozart sucked then!
Also, I’m a big Schoenberg fan and have read a lot about him, and I have no information to contradict what people have written in this thread so far. He apparently was never all that great on any instrument.
I’ve been trying to think about other composers. In the case of songwriters, I think there is a wide range of situations. You have someone like John Lennon, who could not read music and wasn’t really very good at guitar or piano, to people who can’t read or write music but are virtuosos to people who are not very good at an instrument but who can read and write music.
I’ve heard composers say that working without a keyboard results in more interesting music because you’re not limiting yourself to what your fingers can do. I remember an interview with a film composer (I forget who it was) who said that if you compose at the piano you have “your ten dumb fingers thinking for you.”
I once came up with a pet theory that composers who don’t play the piano particularly well, but work at a keyboard anyway, tend to write the most complex music. They’re adding a lot of inner voices and other detail because they’re not concerned about being able to play it themselves, and they’re using the piano to work out unusual sonorities that are difficult to “hear” mentally. Of course I have little or no evidence for this pet theory other than pointing weakly at Schoenberg.
I’d say the opposite. Plays great, but wrote shit. 
If you were including modern composers of electronic music, it’s probably not that unusual at all. Playing keyboards, at least a little bit, is a seriously useful skill when arranging MIDI parts, but it’s not essential.
With more traditional composers, it’s going to be harder to find really clear examples. If you’re going to put in the time to learn musical notation and develop a deep understanding of theory, learning an instrument to some degree is not only very little extra effort, but is actually going to make the learning process much easier. At the very least, you need a good understanding of the capabilities and tonal characteristics of the instruments for which you are writing.
There are examples of composers who were considered poor instrumentalists, both by the standards of their rivals and by the standards of the top professional musicians they were writing for. I would guess that your average person in the street would have considered them exceptionally capable though. It’s all relative.
(edited for clarity/grammar)
On the other hand (and it looks like you cover a variation of this in your next paragraph), you can have “happy accidents” if you let your fingers go and play note combinations you wouldn’t have imagined if you were just going by “what’s in your head.” It really can go either way.
Yes, that’s very true—I remember a Leonard Bernstein essay (or maybe it was an interview) where he describes that exact experience—getting stuck while composing, noodling at the piano, and randomly hitting on something that works.
I was going to mention this. Any composer of electronic music or who even just has a sequencer in their tool kit will be able to easily write music that they can’t play themselves and that in some cases no one could play. But in those cases their musicianship is judged more on their composing and producing ability rather than their ability to play an instrument.
I agree totally, a perfectly valid definition of musicianship could include such things. If we use that definition here though, the idea of a genius composer being a lousy musician seems impossible by definition.
It also depends on how you define instrument. If a Hip Hop producer is using a drum machine or sampling software like Fruity Loops to lay down a beat that they sell to a rapper and shape into a track, I would say that beat-producer is a musician who is a virtuoso at…hmm…beat creation? A specific program that they treat like an instrument to play and compose on? I respect their musicianship, but there’s no classical training, unless “classical” is defined as knowing what to do with the Amen Break and The Funky Drummer 
And distinguishing between composing and songwriting is key. Many untrained songwriters are meh musicians. The concept of envisioning and manipulating deep harmonic structures, as called for by composing, seems it would require an extensive education.
Before the late eighteenth century, anyone we know today as a composer would most likely have been employed by the church, some nobleman or a town council as responsible for performance as well as composing - Bach, Haydn, Mozart, Telemann and their predecessors, and so on.
ok you are so wrong on this:
Actual footage of Beethoven composing.
I’ll go with perfectly ok with him as a composer and musician, except for his solos, which I found, for the most part, tedious.
If we’re letting pop (and variants) creep into this…
Daniel Johnston, who while it may be argued about his “outsider” composing abilities, never bothered to learn the guitar (who some criticised as an affectation, considering he was always a very capable pianist).