Speech does not come with subtitles, so yes, we sometimes elide the "a" in "are" just by the way we talk. If you transcribe speech, then you have the choice of how to represent that. In any case, "there're" is acceptable.
There're a lot of people here.
There're some books on the shelf.
This contraction looks more awkward written down than it sounds in speech. If you are transcribing what someone said, you need not contract this - I would write it down as "There are..." unless I wanted to show that this is very casual speech.
There are a lot of people here; they're all hoping for a bargain.