It's pretty challenging to find horse property rentals in the surrounding areas these days. But here's the neighborhoods you should check out.
30-45 minutes from Seattle (with no traffic) are Redmond, Woodinville, Bothell and Renton.
You could also look in Issaquah and Snohomish, though with traffic they can take a little bit longer to get to. There are a ton of boarding barns, most have indoors and a few have access to trails also. Prices are usually between $500 and $600 for full care (you can find ones without an indoor for lower and there's a ton of fancy barns that are higher).
If you find a barn and want to know about it, send me a message! I've either boarded at them, taken a lesson or visited. :) Good luck!!