I bought the batteries from ampul.eu website.
Totally recommend purchasing from this website. Fast delivery and I also got a bonus in the package.
It came in from Czech Republic so it is suitable for the EU market.
I don't know about the US market though. Anyhow the size should be a max of 40x30x4/5 ... ideally a 35x30x4 would be the best option IMO.
Battery link: https://www.ampul.eu/en/battery/4140...

And the link from Ifixit where they show you how to dismantle the bot.
Ifixit link: https://www.ifixit.com/Guide/Anki+Vec...

Also this is a very thorough research manual done on vector robot by Randall Maas. Totally recommend it if you want to play around with vector
Manual link: https://randym32.github.io/Anki.Vecto...

If you have any questions, feel free to ask them in the comment section. Thanks for watching!

00:00 Initial battery timelapse
02:42 Disassembly
15:09 Soldering
24:03 Head Gear Assembly Issue
26:55 Reassembly
44:27 New battery timelapse
48:50 Outro