I've just successfully (apparently) compiled a TeX document on
sharelatex.com, using a .bib file as input.
But to submit to the arXiv, I require a .bbl file.
(I don't presently have access to TeXShop or similar programs.)
How might I get it? It seems that I get an .aux file with the output,
but no indication of a .bbl file.
Do I just need to wait until I have access to TeXShop?

.bblafter a successful run. You can view the.logfiles and there is an option on the bottom right (more or less) to download auxiliary files, there you can choose the.bbl. – moewe Dec 22 '15 at 18:13.bbl- of course that only appears if you have a bibliography set-up that needs.bblfiles). – moewe Dec 22 '15 at 18:16