How do I evict a groundhog living under my porch without trapping or poison?

Essentially, you have two options.

Option 1. Hazing also known as harassment.
Harassment is just continually back filling the hole until the animal gets tired of digging out and leaves, but it could take weeks. Hazing Methods

Don’t waste your time with repellents such as mothballs and ammonia as they will either be ineffective or will irritate you more than the woodchuck.

Option 2. One-way doors.
One way doors can work provided you trench screen your porch area and install a one way door over the opening. Use caution when young are present, typically prior to July. I don’t recommend the inexperienced to try one-way doors as too many things can go wrong. To trench screen, (watch out for buried wires and pipes etc.) you need to have screening above ground attached to the porch wall (1/4 inch mesh will work, galvanized is better). Then go down two inches under the soil and make a 90 degree angle out away from the porch wall at least 18 inches. This way, when the chuck meets the wall, he wants to dig straight down to get under it. But when he does he is trying to dig through mesh which he is standing on. Of course you can bury the mesh deeper if you wish and make is larger as well. Exclusion

Additional Resources (Online)
Woodchuck Control Methods