Control strong DC motor with microcontroller
You answered your own question; you'll need an H-Bridge. Here's a DigiKey listing of H-bridges that are rated for at least 5A, and for supply voltages at least to 24V, sorted by price ascending. The Infineon BTM7700G in that list is relatively inexpensive and exceeds your requirements by a healthy safety margin -- and, since it's a 1.27mm pitch SO package, it's extremely easy to solder.