Saeco Vienna machine...after brewing a couple of cups of coffee, water leaks from somewhere and comes out underneath the front grilled tray and onto the counter. Can I do the repair with the proper parts?

Depending on your ability to do some technical and minutious manual work, unmounting and remounting the machine is feasable.

Concerning the leaking, what I've seen most is a leak from the seal under the water tank. Any leaks coming form there would occur even though no coffee would be dispensed. (i.e: Idle mode)

If leak is ONLY occurring when brewing cofee, I would check the following:

1- Check if the water inlet on the back of the brew unit is filled with residue (the inlet looks like the end of a smoke pipe)

2-Open the machine, take out the interior, and verify the water tubes coming in and out of the boiler. They both should be sealed with 2 small o-rings. After time, o-ring tend to expand under heat and may start to leak.

Hope this helps