There is a trap club (Ogden) at the north end of town, and one at Brigham City, both near I-15. In addition there are the Salt Lake club and Magna in/near Salt Lake City, and Spanish Fork just south of Provo. Also Pocatello about 120 miles north on I-15. All throw ATA shoots. You'll have great competitions within a reasonable drive.
As Larry ("Pocatello") mentioned, there's a trap club (Golden Spike) in Brigham just south of the I-15 Forrest St. exit. That would be the closest. They have practice on Thursday evenings and Sunday afternoons. They usually have two ATA shoots, one in May or early June and one on Labor Day weekend.
Ogden Gun Club is actually in Willard. It's located just north of the Willard Bay South Marina entrance (2000 W 8700 S). The nearest I-15 exit is 352. They shoot on Wednesday evenings and Sunday mornings. They usually have 4 ATA shoots. A one day shoot in late February/early March, a three day shoot in April (the Northern Utah Open) and a couple of two day shoots in the summer and fall.
The link above to the USTA's site will provide you with a little more info.