Yes, anyone who has had their property seized pursuant to a search warrant can appeal for relief. Especially an innocent party might contact the law enforcement agency that seized the property, and/or the government attorney's office handling the case for which the property was seized.
Although I am not aware of formal rules requiring the state to compensate third parties for the costs imposed on them by judicial processes, I have seen police departments and distract attorney offices make accommodations, including payments, to defray those. (Perhaps this is motivated by nothing more than an interest in maintaining positive community relations and avoiding bad publicity.)
The formal process to obtain relief is to file a Motion to Return Property. Following is the text from FRCP 41(g), but state rules I have seen are virtually identical:
(g) Motion to Return Property. A person aggrieved by an unlawful
search and seizure of property or by the deprivation of property may
move for the property's return. The motion must be filed in the
district where the property was seized. The court must receive
evidence on any factual issue necessary to decide the motion. If it
grants the motion, the court must return the property to the movant,
but may impose reasonable conditions to protect access to the property
and its use in later proceedings.
Finally, it is possible that private vehicle insurance will cover costs associated with being deprived of a car seized as evidence. (The closest analogy I know for certain: If a law enforcement agency causes damage to property, many homeowner and renter insurance policies will cover that damage under their "casualty" clause.)
