Well, I use MySQL professionally. So my understanding of it is with a business infrastructure supporting it.
You can install MySQL locally and run it yourself fine. But it's not one consolidated file. Each table in the db is a file, and there are lots of other supporting files for the db. It's not something you can "host" through free file sharing sites. If you wanted an actual, accessible-from-anywhere-but-protected setup, and you're not capable of doing that yourself from your own machine, you'd maybe want it hosted by the Amazon or Google cloud. But that also expects a certain level of technical sophistication on your end to make it work. (And, ya know, having someone else host it for you costs you $$ every month.)
What you might be able to get away with though is....
Both of you install MySQL and SQLYog.
You export a copy of the database as a SQL dump file, zip it, put it on Google drive.
He downloads it, and executes it on an empty copy of the database using SQLYog's import function so it recreates everything. Then he can browse stuff, make changes, etc...
Then when he's ready to pass it back to you, he does a SQL dump, zips it, yadda yadda....
And on your end, you truncate the entire database, then restore it from his SQL backup.
It's not as elegant or smooth as you both having live access to the same DB. But it would allow you to swap copies back and forth between each other. I can't imagine the file size footprint of your work is very large (I usually deal with database dump files of 50+ gigs or larger) unless you're storing image data in your db, so it should make for a small file that takes no real time to download and import. The biggest limitation is that you can only really have one working copy at a time since there's no easy way to merge your changes with each other. On the plus side, SQLYog essentially allows you to view the db as a spreadsheet and edit that way. So they wouldn't have to really learn any query language.
And if you get more sophisticated with the SQLYog, you can do things like only export the tables you've actually changed and just truncate --> reimport the tables you need rather than doing the entire db. Which, again, I can't imagine is large enough for it to be that complex yet.
TLDR: I don't think there's a way for your brother to have DB access without them actually installing something on their end, or you paying money. If those are lines that can't be crossed then you're probably just better served by Google Doc spreadsheets. And if you go the route I suggest, then it'd probably just be better to have them install MS Access since you're already familiar with it and can help them. I haven't used MS Access that much but I kind recall its db browser is shit.