Kim and Jim’s Wedding.
or
Kim’s and Jim’s Wedding.
Kim and Jim are the couple, so I’m inclined to choose the first option.
To me the second sentance would infer that there were two weddings.
Crap,
Sentence
Why is it that everytime I ask a question about spelling or grammar, I screw something up in the post?
a) the word is sentence, not sentance
b) the word you’re looking for is imply, not infer
c) The wedding is that of Kim and Jim, therefore Kim and Jim’s wedding is just fine.
My take: the latter is strictly correct, and should be used when writing it - otherwise you’re talking about two separate things: “Kim”, and “Jim’s wedding”.
If the couple are generally considered an “item”, you might get away with “Kim-and-Jim’s wedding”, but it might look a little twee.
However, colloquially, the former would sound weird spoken aloud, so the latter would be more appropriate.
Well I aint no grammarian, so you may not want to listen to me. The first sounds correct to me, but I suspect the second is actually correct (in the fifth grader portion of my mind a teacher is explaining how the sentence should make sense if you remove either “Kim’s” of “Jim’s”, i.e., you wouldn’t say “Kim wedding”). I anxiously await someone who is more smarter to illuminate me 
<donning copyeditor hat>
I beg to differ. The first one is most definitely correct.
See “Compound Possessives” about halfway down the page for a simple explanation.
That’s how you determine case forms of pronouns, not possessives:
“The house belongs to John and me.” You can take out “John and” to see that “me” and not “I” is correct.
In the OP’s sample, “Kim and Jim” is the compound unit that takes the possessive.
I’d prefer the first sentence. Nobody in their right mind would think that “Kim and Jim’s wedding” would mean anything other than “the wedding of Kim and Jim.” You’re talking about one wedding.
If, on the other hand, you were talking about two separate events, e.g. Kim and Jim are maybe your siblings, and had two separate weddings to two other people – then, you’d say “Kim’s and Jim’s weddings (plural).”
But I’m not sure whether this would hold if they had a double ceremony.
And yes, the word is “imply,” not “infer.”
I believe it’s called Gaudere’s Law 
Well, whaddya expect from a fifth grader? 
Thanks for the info, Scarlett.
Nope. Gaudere’s Law is when you’re posting to correct someone else’s spelling, you’ll make a spelling error of your own. Just asking a spelling or grammar question doesn’t trigger the Law.
From the OED:
Infer, v, 4 To lead to (something) as a conclusion; to involve as a consequence; to imply …
Yes, the prescrptivists have tried to straitjacket the two words (and, admittedly, it’s a useful distinction) but historically both words have been interchangeable for centuries (first cite for infer in this sense from the OED is 1530.)
Yeah… I’ve done that too! 
And perhaps another “prescriptivist” will be along to correct your spelling.
In a computer programming sense, I would code it as:
(Kim and Jim)'s Wedding.
Making clear that the 's possessive applies jointly to Kim and Jim.
John and Mary are a married couple who jointly own a house and two cars. John and Mary’s cars are parked in their garage.
Sam and Stella are coworkers in the office, whose only out-of-office contact is at the annual office Christmas party. Sam’s and Stella’s cars are in the parking lot adjacent to the office.
Yes, formally the first is absolutely correct. An apostrophe-s need not simply take the word it’s attached to–it can take a noun phrase.
If we need even more citations backing this up, here’s the American Heritage Dictionary Book of English Usage, 1996:
But since the distinction is useful, they are not truly interchangeable in the minds of some. And, according to old reliable Strunk and White, it’s at least a matter of style. If nothing else, it allows for some weaseling in GD.