Don't let the word "permit" scare you. For an elopement, it's usually straightforward:
The marriage license. Get one from an Arizona Superior Court Clerk (Yavapai, Coconino or Maricopa County are nearest). It's about $83, has no residency requirement, is valid statewide for a year, and there's no waiting period. You'll need two witnesses over 18 — and if it's just the two of you, you can simply ask friendly hikers on the trail to sign. Your officiant files it afterward.
A permit often isn't required for a small, simple ceremony on Forest Service land with no setups — no arch, no chairs, no décor. You will need a Special Use Permit if you add those things, bring vendors, use amplified sound, or have a larger group. Wilderness areas (like Devil's Bridge or the top of Cathedral Rock) don't allow ceremonies at all, and state-park sites like Crescent Moon require a permit and fee.
Leave No Trace. A Red Rock Pass covers trailhead parking, and please skip the rice, petals and confetti — even the "biodegradable" kind.
When in doubt, confirm with the Red Rock Ranger District, or let a local photographer or venue handle the paperwork for you.