For just coarse adjust, a geared dc motor – window or wiper motor – sounds the way to go.
If you want more precision, take a look at (used) NMRV-030 worm gearboxes and NEMA-23 stepper motors under microcontroller control.
I got one off eBay for a satellite tracker antenna elevation (Alaskan Arrow II). (I don't do satellite stuff, but a friend does.)
If I were in your position and the crank can be used for finer adjust also, me being in Europe, I'd look at e.g. rattm NMRV-030 on ebay, maybe 20:1, and a dual 8mm shaft (for hand crank on the butt end just in case) NEMA 23 stepper motor (say, 23HS9440B-26), and a TB6600 stepper driver, and just about any microcontroller for generating the pulses. I'd also use a two-step linear regulator from whatever voltage you use on the TB6600 to get the 5V to the microcontroller, to avoid motor noise causing glitches. At 3A or so, a 24VDC supply at 4A or more should work. The only trick on the microcontroller is to use ramping for both start and stop, to reduce wear on the gearbox. If the stepper+gearbox has much more torque than needed, then one can assume no steps will be lost, and count the steps to calculate the distance moved. Or you could add a 5-micron scale, and read it using the microcontroller as well.