I’ve seen it enough to think it likely that you’ve seen it, probably many times over. From what cultural source do people write “welp” where you’d expect “well” in the sense of “Well, folks, it’s like this, you see”. Or, um, “Welp, folks, it’s like this, you see…”
??
First time I saw it aside from OP is via search finding a BBs with an OP beginning with it.
WAG A simple case of tangle fingers/nerve pulses producing a typo. “p” and “l” are diagonally adjacent on the beykoard! (deliberate) 
Possibly a Souther Appalachian idom handed down but I am not aware of such usage.
When you say the word “Well” you finish by sort of pursing your lips as you close them, the last exhalation of the word coming just before you close your lips. If you just exhale a little more as you close your lips you’re basically making a solo “P” sound (sounds like “puh”) and it’s easy to see how that turns “Well” into “Welp”. I notice that I tend to do this if I’m saying “Well” as a drawn-out word.
Dunno if it’s a regionalism, I kind of associate it with a country, twangy kind of accent but I’m a Berkeley boy out here in central CA and I do it too.
Ditto @ Valgard
Welp, I gotta say ditto to Sage Rat’s ditto of Valgard’s post.
I don’t purse my lips after saying ‘well’. But my idea is similar to yours. ‘Well’ gives the impression that what the speaker is about to say is well thought-out, and that he is thinking of a way of phrasing his statement such that these deep thoughts of his will be conveyed clearly and concisely. If there will be a significant pause, then it wouldn’t do for the speaker to stand there with his lips parted and the tip of his tongue behind his teeth. So he shuts his mouth. Perhaps someone has gotten into the habit of starting sentences with ‘Well,’ and pausing long enough to close his lips. He still closes his lips after saying ‘well’, but then keeps talking – resulting in the asperation Valgard mentions.
“Welp” is often used in a summing-up, when the result is already known, such as in “Welp, this puppy is toast.” Whelp, of course, is a puppy or the act of giving birth to a puppy.
Aw, fooey. That’s a stretch, isn’t it.? It seemed like a good idea for a minute or two.
Not to add fuel to the fire, but what about :
"Yep" and "Nope"
And “Yep” can go to “Yop” as well.