Hi there!
Some companies have restrictions on allowing children in their trucks, so you might want to check with Budget directly.
I would also be worried about airbags in the truck. It's very possible that a rental truck like that is an older one (I don't know how often they get new ones, and I imagine they last for a while). That means if there are airbags, they might be an older generation that was very dangerous to young children, even forward-facing ones. (Obviously a rear-facing child should never go in front of any active airbag.)
I imagine a truck like that also wouldn't have the safety amenities of side airbags and stuff that modern passenger vehicles have. Plus, there's the fact that children are safer in back seats.
So...just stuff for you to think about.
We recently moved from California to Illinois. My husband drove the truck and I drove the van with all three of our kids. Other than a few fits here and there, it actually went much better than I imagined. I highly recommend portable DVD players and iPads if you have them
Good luck!