Is Vico Esquense a good base for visiting Amalfi Coast?

Before you rule out Sorrento, you might try emailing, or calling a hotel,etc, that interests you. I've had the experience of finding places booked on their own website and the booking sites, but they had a room after personal contact.
I don't know anything about Vico Esquense other than that we just passed through it last week several times in a bus and taxi. Like most small towns in Italy, all the roads are one way each direction, so traffic moves very slowly. You don't say if you will be using public transportation or driving. If using public transport, I would research if you can reach the Amalfi Coast by public transportation without going thru Sorrento which would add time to the trip. You would probably get the SITA bus to Sorrento in order to catch the ferry to Capri. I would check to see if the Circumvesuviana train has a stop near there to get to Pompeii.
Good luck! We spent 4 nights in Sorrento, then 1 night in Capri. We actually wish we had spent more time in Capri. There is a fair amount of hiking, gorgeous views, and after the crowds left, it was very relaxing.

Might also want to check out Ravello or Maiori on the south coast, as I've heard they both are spectacular. Or the small city of Salerno just off the peninsula, which is logistically perfect for daytrips almost anywhere in the vicinity. We based there for 5 nights a year ago March and loved it.

Given your preferences, you might prefer the Amalfi coast itself. Positano, Praiano, Amalfi, etc. are all excellent choices and they may have still accommodations within your budget. There are boats to Capri from both Positano and Amalfi.
Also don't know much about Vico Equense. I also just passed through it. I suppose it's ok too. To go to the Amalfi coast it's actually faster if you catch the bus along the way at the Meta di Sorrento Circumvesuviana station.

We stayed at the Hotel Panorama in Maiori - it was lovely and inexpensive.

Talked with someone at breakfast today who has stayed, and is returning to stay in Vico Equense. He said it is a lovely little town with few tourists and good restaurants. You do have to go to Sorrento for connections to the Amalfi Coast, but it wasn't too long, he said.