Yaris starts then it wont start.

[ilikerice]

My gut makes me think it might be something like the fuel pump relay

im gonna second this one. if the fuel pump relay coil side gets hot and worn, it will have a hard time producing a current thru the coil to get a good enough magnet to pull (or push) the switching side of the relay. when the relay is cooled down enough, current can run easily thru the coil and create a stronger magnetic coil.

we have a problem like this with most little civics with thier main relays, the main relay is also the fuel pump relay.
if you wanted to start with the cheapest part first and go from there, i think the fuel pump relay is your cheapest way to start off with