Do we need train reservations from Paris to Vernon?

Not for either.

The RER to Versailles runs about every 15 minutes.. and costs 4,10 euros each., You can buy them from a machine(n if you roll the order bar you can find english) or you can get them at any manned kiosk in a metro or RER station, and you can buy them anytime you like ( they are not date or time sensitive) so buy them any time you see a manned window.. You want an "aller retour Versailles Chateau or Versailles Rive Gauche" ( not Versailles Chantiers.. that train goes to Versailles but that station is 25 minute walk to Chateau other one is less then 10.

Not only do you not need reservations for either of those trains, they aren't available. The train to Vernon is an open seating Regional, the RER is is like a fast Métro train with 2 decks and harsh seats, first come first served, stand if full.

I think you can get the tickets for the gardens on line -- and use whenever i.e. undated. At any rate, very good idea to have them in hand when you arrive. They key to success in crowded times at Giverny is to step lively to the bus when you get off the train in Vernon. It does fill up and leave people standing and when it did it when we were there, there was no other bus in sight. You buy tickets on the bus. Report to the bus stop at Giverny 20 minutes before the return time to be sure of getting a seat or at least a spot back.

For sure, get your ticket to Monet's garden in advance. I bought mine online and with tickets in hand I walked right up to the front of the line of people buying tickets and was immediately let in. Enjoy your visit, it is so beautiful!