Is "there're" (similar to "there's") a correct contraction?
Solution 1:
There're is common in speech, at least in certain dialects, but you'll rarely see it written. If I were being pedantic, I'd advise you to use there are in your example, because there is is definitely wrong, so there's could be considered wrong as well. But a huge number of English speakers, even those that are well-educated, use there's universally, regardless of the number of the noun in question, so you will probably not receive any odd looks for saying or writing there's, and if you do, just cite the fact that it can't be incorrect if a majority of people use it. As for me (a native New Englander), I use both, but may use there's in place of there're if I'm speaking quickly.
Solution 2:
I don't think "there're" is ever going to fly -- it's not so much a contraction as a simple elision. The only thing being dropped is a glottal stop, which isn't a "real" sound in English.
From a strict prescriptivist grammar and usage standpoint, "there's" used with a plural is wrong. But in spoken language (which is the real language, squiggles on pages and screens are no more than an approximate rendering) we need to be careful with prescriptivist tendencies. It may offend the grammarian's ear, but the fact that a very large number of native speakers -- likely a preponderance of them -- make exactly the same "mistake" indicates that there is something else going on.
Remember that the rules of English, as we received them in school, are only an approximation of the real rules of the language, and that many of those rules were imposed in the 18th and 19th centuries by well-meaning scholars who aimed to make English a respectable, consistent and properly-documented language. It has never been such.
Solution 3:
Here's my guess:
Contracting "there is" to "there's" usefully reduces the number of syllables.
Contracting "there are" to "there're" wouldn't. So it isn't done.
If I reach for a shorter way to say "There are some in the closet", my mind produces "They are in the closet" (favouring utility over accuracy).
Obviously it would help to know how you pronounce each of "there are" and your "there're". Perhaps we should both study IPA for a few minutes?