Have you contacted Customer Support through this form and asked for the Pass Cover Number?

https://eurail.zendesk.com/hc/en-001/requests/new

Customer Support is currently overloaded with requests so you will probably have to be patient to get help. If you are currently travelling or will be travelling in the next weeks you should inform Customer Support about that.

For most routes there are different possibilities to make reservations and some don't require the Pass Cover Number.

I did contact customer support but noticed that some people were getting quick responses to this chat forum. Thank you for your suggestion. I wasn’t aware of alternative options for the route I am taking. However, I will look into it immediately. Much appreciated!

Here are guidelines on how to make reservations.

https://community.eurail.com/train-connections-reservations-47/how-to-get-reservations-105

If you're already in France, just go to the ticket office.

If you're in a hurry, just call SNCF. Press #85 for English, no booking fee, reservations will be e-mailed.

For TGV in France, calling SNCF via the Hotline is simpler, faster and more helpful than any websites related to Interrail/Eurail or SNCF, if no tickets are left via Interrail and you have a mobile pass and can’t book via travel.b-europe.com.

I was able to reserve my seats within 15-20 minutes.