(Disclaimer: I'm a newbie) I've read various numbers up to about a month. Four months might be stretching it.

Alrighty. I think I'll probably end up freezing it then. I've been canning for a long time but this was my first time with tomatoes. Just assumed they'd be so acidic it'd be fine! Thankfully I googled it before I gave my whole family botulism. :o)

"My assumption is that with the proper sealing (no air coming in) and the temperature (anything in there cannot grow)"

Sorry but isn't a valid assumption. First, because there is air still trapped in the jar and the acidity level will also quickly rise allowing for bacterial and fungal growth to occur - neither need much air to do so.

Storage life for unacidifed, low acid foods in the fridge can be as little as 3 days as that is when listeria begins to grow in it.

You must freeze your sauce. If less than 24 hours old you can put it directly into the freezer. If more than 24 hours old it must be fully reheated to boiling and then packaged and frozen.

The only other alternative is pressure canning it.

For future reference, tomatoes alone are borderline low-acid and require additional added acid. So there is no way they can cover their own pH much less that of onions and basil. If you aren't familiar with current guidelines I'd suggest some reading at NCHFP.

http://nchfp.uga.edu/

Dave