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This is old but it didn't get answered at all. Hopefully OP found the info elsewhere but...

Yes! In typical 'Murican fashion we have TWO sanctioning bodies NRA and CMP

Service Rifle here is somewhat similar to UK:
We use semi-auto rifles(sometimes modified in ban states)

Across the Course
600 slow prone
300 rapid prone with reload
200 rapid fire sitting(or kneeling but rarely chosen) with reload
200  standing

Many small variations exist for smaller ranges, different classes of rifle, etc

Prone
All shots slow prone at midrange(300,500,600) or long range (800,900,1000)

The Sierra Vista Shooting Range has a match called Civilian Battle Rifle.  Here is a description from the Range's website:

4th Sunday (even months) - Civilian Battle Rifle

Targets are a human sized paper silhouette.

Course of fire:
• 10 shots lying down in 90 seconds from 200 yds
• 10 shots kneeling/sitting in 90 seconds at 120 yds
• 10 shots standing in 90 seconds at 80 yds

Rifles used can be any rifle scoped or using iron sights which you may use in case of an invasion by foreign powers to protect yourself/country. This is a short (thirty rounds) match that tests the shooter's skill more than the rifle/ammo capabilities. Each stage, by the way, requires a magazine change somewhere within the 10 round course of fire. This match emphasizes the ability to maintain a sustained rate of accurate fire, rather than single shots to target. Use of a sling is allowed, and makes shooting easier.

I have not shot at this competition, but I did shoot a charity event with similar rules and it was fun.

Have you found the matches here in AZ that you're looking for?

XTC and FPR