Hi @JoshuaDahl. Welcome to GoDaddy Community! I'm not familiar with libsodium, but I would guess you probably would need at least a VPS with us to be able to run it. However, it may be possible. If you can post more information about the errors you're running into, someone may be able to suggest a way forward for you.
libsodium is available even in the most basic packages on Godaddy Cpanels.
If you've had a cpanel account for a long time, it may require you contacting support to upgrade, if you have a cpanel account you can enable libsodium via the php version in cpanel.