Does an airport need to have at least one instrument approach that I can use in order to fly IFR to it?
Under Part 91, you can file IFR to any point in space, including an airport without an instrument approach. But that doesn't mean you'll be able to land there.

FAR 91.175(a): "Unless otherwise authorized by the FAA, when it is necessary to use an instrument approach to a civil airport, each person operating an aircraft must use a standard instrument approach procedure prescribed in part 97.."

What ATC can do is clear you to that point in space at their lowest vectoring altitude there. If, when you arrive, the conditions are VFR, you can cancel your IFR and land. If not, you have to go somewhere else, possibly the alternate airport you would be required to have filed in your flight plan. And it's that alternate airport that MUST not only have instrument approaches, but have weather forecasts and reports high enough to meet the alternate minimum requirements of 91.169(c).