Hi Anne,
In theory it is possible to replace just the charging port, but the problem is it takes microsoldering skills to do so. I consider myself very skilled at home repairs, but I don't think I'd tackle that one myself. The charging port is soldered to the motherboard, so unlike iPhones and a lot of other phones, you can't just replace a part that has the charging port on it.
So you really have about two options; either take the phone to a qualified repair shop that has experience in microsoldering and have them replace the port on the motherboard, or replace the entire motherboard.
You can buy used ones off eBay, but you do have to be careful to verify that they're both Google and carrier unlocked. I ran across one on AliExpress that looks like it would do the job for about $100 USD.
[link|https://www.aliexpress.com/item/3256804190257558.html|For Google Pixel 5 Motherboard Main Board Work Well Unlocked 100% Original 128gb - Motherboards - AliExpress|new_window=true]
Of course, you'll need instructions on how to replace it, and iFixit has your back on that one.
[guide|140529|Google Pixel 5 Motherboard Replacement - iFixit Repair Guide|new_window=true]
It certainly wouldn't hurt to call around to some local shops or hit up some on the internet and get an idea what it would cost; who knows, you might get lucky and find someone who can do it for a reasonable price.
Best of luck with your phone; let us know how it goes!