How far in advance can you book train tickets from Paris to Avignon? Travelling in late May.

Around three months.

Depends on the travel date and when SNCF is ready to sell them. Usually 3 or 4 months in advance, but summer dates (through the end of August) tend to be released in a couple of big batches, sometimes 6 months out. You can sign up for email alerts at https://en.oui.sncf/en/deals/opening-reservations.

For late May, TGV Ouigo fares are already for sale: https://ventes.ouigo.com/en-GB. These budget versions of the TGV fast trains work somewhat like a no-frills airline (e.g. they limit luggage size and seat selection, require early check-in, sell online only, and don't accept rail passes).

The SNCF will contact you when tickets go on sale for the routes of your choice. This is not just a low fare alert (as in the link in the above post), it alerts you when any ticket is available for sale on your chosen route.

Alert Résa is in French only but you should be able to input your departure station, arrival station, desired date, and your email address for eventual notification (usually about 3 days in advance of sales opening).

The SCNF website with the email alert sign-up for trains not yet bookable was in English. Laura: Thanks for the link!