My 1978 FIAT Spider has a mechanical fuel pump. Is it possible to install an electric fuel pump near the tank and have both of them operating?

Your car is not wired for an electric fuel pump, but if you looked at a wiring diagram for one of the years that did, you would be able to wire in all the relays and safety controls just like the factory did!

Would the electric fuel pump from an earlier Spider be suitable

It would be, but they are so expensive we usually convert the electrical pump car to a mechanical pump! Getting a good quality electric fuel pump these days can be a challenge. I spent over $150.00 for a good quality one for my truck, I got tired of the $85.00 units leaving me stuck!

or is there a better option.

The mechanical is the best option.

Why do I ask? I have been having some problems with hard starting. If my car sits for several hours its acts like there is no fuel in the fuel lines or carburetor. A lot of cranking will get things running finally.

I suspect you have a problem with the carburetor. Either the float level is too low, or it is leaking the fuel out of the float bowl while sitting. Installing an electric fuel pump may allow you to start the car, but will not solve your problem - it will only mask it.

I noticed in the shop manual there should be check valves in the supply and return lines to the tank.

Yes, but they had a tendency to leak! Fiat even had a recall to get them out of the system!

My car does not have these right now.

Good!

I have also had some minor problems with 'vapor lock' on warmer days.

Fiats don't have vapor lock problems. I feel your problem is someplace else, and I suspect the carburetor.

Thanks

Glenn Swarner