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Would it be possible to have an electric motor that uses an electromagnet to force a piston up and down to turn a shaft, similarly to an internal combustion engine, but without burning fuel?
Is there a name for this kind of motor?
Would there be any advantages of this style of motor compared to other electric motors?
Linear electric motors exist, and are used in applications where you want a linear motion. As Solar Mike said, using a linear motor to drive a crank is pointless.
Every time you convert energy from one form to another, the conversion is not 100%, there is some loss. For example: electrical energy to mechanical energy (motor), chemical energy to mechanical energy (internal combustion engine), linear mechanical energy to rotary mechanical energy (piston/connecting rod/crank). Many systems will have multiple different conversion steps, and the losses can really add up. Therefore, you are always better off doing as few conversions as possible. If you want linear mechanical energy, start with that in the first place. If you want rotary mechanical energy, start with that in the first place. Only do conversion if you absolutely have to.
A standard electric motor turns the shaft directly (ok we may or may not need gearing) which keeps the efficiency high. As soon as you add pistons, connecting rods, a crank you reduce the efficiency and add weight. The piston / crank is also used in pumps.
Typical electromagnetic actuators create this kind of motion when you want linear movement in few (2-3) large, discrete steps, e.g. a latch mechanism (for door etc) or other back-and-forth movement, more or less rapid (electric bell, membrane pumps, relays) usually employing a spring for one position, electromagnet for one or two more - you'll rarely see ones for more discrete steps.
It would be possible to employ these like pistons of a combustion engine, but the efficiency would be inferior - they don't provide nearly as much force/torque (for the same mass and wattage) as classic (rotary) electric motors.
It would be possible.
If it exists, its uncommon.
For any application it might serve, it would have to compete with solenoids, where a spring accomplishes the return stroke. In my youth I worked for a company that used a solenoid and ratchet to drive the sprocket wheel in a paper tape reader. It had respectable speed, but ear protection (should have been) recommended.
