Is there a serial input buffer in the LC, like there is in the arduino? If so, how big is it?
Yes, there is. It's defined in
serial1.c. The default size is 63 bytes. Due to the way the buffer works, 1 byte can't ever be used, so a 64 byte array gives 63 bytes of actual buffering. The buffers in Arduino's code work pretty much the same way.
If you really need a bigger buffer, you can edit this to a larger number. But larger buffer sizes aren't the answer to all problems.
I've tried sending over 300 bytes and my programs still load and run just fine, so I'm assuming it's at least as large as that
If your program uses Serial1.available() and Serial1.read() to fetch the data as it's arriving, odds are you're grabbing the data quickly enough that the buffer never gets full.
If you want to know more, you could try printing the actual number Serial1.available() returns, instead of just checking whether it's zero. Everything you add to your program, like printing extra info, will take away some of the time you can use to read data out of the buffer. But if that printing shows the available() is never more than a few bytes, you can feel pretty confident things are going well.
If you're really paranoid, you could add some code to keep track of the largest number available() gives, perhaps printing that every minute or two and then reset it. Then you can monitor a "worst case", and usually simple comparisons and just storing stuff into variables is very quick.