nilsfbx For the future, you need to install WhatsApp AFTER installing Play Services + Google Drive and after logging into your google account. Backups will work normally after that. Sometimes this means re-installing Google apps and logging back in. (If you don't care about Google Drive backups, then WhatsApp works perfectly fine without Play Services).
Similar issues will happen with other apps that expect Google apps to be pre-installed, such as Google Contacts. Kind of wish this was mentioned somewhere so other GrapheneOS users don't keep running into these problems.
Since this means you'll need to re-install WhatsApp in order to get Google Drive backups to work, the issue of how to backup your current history still stands. I'll leave it to someone else to see how to do a local backup. I think it's still possible, but I haven't had luck in doing it in a few years, even on stock.
If you can't get local backups to work, if you have a separate device, you can try first transferring your message history to that device. Then either set up backups on that device or transfer it back to your current device after logging into your google account like I described earlier.
Also, hopefully it doesn't come to this, but in case you're unable to get the backup working, WhatsApp allows you to export chat histories for each chat, which will include a .txt file of the messages as well as all of the media for that chat.