Will a standard sedan have enough room for 3 full size suitcases and 3 people

Unless, of course al three people have roller bags or carryons. Then it could get tight.
The other thing to worry about is how the driver will respond.  Most drivers try their best to be accommodating but some will frown upon people forcing too much on, especially if there can be damages to their cars. 

If you call an UberX and realize it was right, make sure to tip. In cash and in the car to avoid negative rating. 

I’d say yes.  If you get a larger full size sedan, like a Toyota Camry, three large suitcases can fit in the trunk.
Even with a smaller sedan (or larger suitcases that don’t all fit in the trunk), technically one of the suitcases can go up front and take up one of the passenger spaces. Let one person sit in the front passenger seat (next to the driver), and have that suitcase be placed between the two people in the back.

I don’t think it’s necessary to call an SUV for that.