Jesi is too small to have car rental, unless there is some tiny local rental company. You will probably need to pick up in Ancona.
Have you asked your relatives?
The Ancona Airport is only 17 km far from Jesi. Budget rental car is there for sure but would expect other providers too.
Have a look at www.autoeurope.com for car rentals. They have a pick up at Ancona airport and Ancona downtown and deal with many companies. We have rented from Ancona airport in the past and it is an easy drive back to Jesi. Also, there is a local rail station at Ancona airport(Falconara).Edited: 23 March 2014, 16:10
I previously forgot to mention the train station by the airport: it is Castelferretti.
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