New arrivals routinely find this out the hard way, having driven fifteen minutes up a canyon with a dog in the back. The rule exists for a straightforward reason — the water in those drainages goes to the tap — and there is no permit that gets round it.
The canyon that does allow dogs is Millcreek, and it operates on a rule worth memorising: above the fee gate, dogs may be off-leash on odd-numbered days of the month and must be leashed on even-numbered ones. Two odd days fall consecutively at the turn of most months, which is the sort of detail that catches people out. There is a fee to drive up, and it is modest.
Emigration Canyon is the other nearby option, and the Bonneville Shoreline Trail runs along the benches above the city for miles, connecting neighbourhoods to open ground without needing a canyon at all. Between those and the city's parks, a Welsh Springer can be genuinely well exercised here. What it cannot do is default to the obvious mountains, and an owner who has not worked that out in advance loses the first month to frustration.
So the practical advice is unusually concrete. Learn which canyons are closed before you need them. Put the odd-day rule for Millcreek somewhere you will actually see it. And treat the Shoreline Trail as the everyday resource rather than the fallback, because it is the one that works every day of the month regardless of the date.