What would happen if Render changes their plans and I lose access to the database? Will I still have access to the last-stored cache on my browser extension and mobile phone?

Yes, the bitwarden client will simply treat it as being offline. You should check the docs on how to migrate to a new server so you can be prepared.

And since I’m running a Rust infrastructure, would it use less of the free plan bandwidth that Render assigns?

No. ~~Bandwidth is up to the network stack to determine, not the programming language. Generally, your app and OS will use as much as avalable unless otherwise throttled.~~

I just looked it up, and their "bandwidth" is not a measure of bandwidth, but a data quota. The answer is still "no" because it's about how much data is transferred in total, which has also little to do with the language in this case. Despite the difference of some negligible amount of bytes of overhead, vaultwarden's limited by the format the database is in. To lower data usage, try reducing how often you automatically sync the clients with your server.

I'm planning to run Vaultwarden on a free instance of render.com, and I wanted to know if this was a good idea? Has anyone over here tried this?

I have not tried this, but i am opinionated: on one hand, self-hosting will always be your most reliable and private option. However, if you have judged other pursuits a more valuable use of your time and mental energy, then it's probably worth the $20/month (or whatever) if and when your server lands in reorganization jail.

The biggest issue would be your privacy, which almost always goes out the door when money comes into the picture.