Knowing absolutely nothing of the Telluride long-term rental market but dealing with a similar style of resort rental market in Summit County, I would be open to going out there whenever you can find a rental that you like that is in your price point.
You may want to start looking around now to just get an idea of rental prices to make sure you are in the ballpark. Get addresses and use mapquest.com or other online map features to see where the rental is in relation to town especially if you will not have a car.
In my opinion, you might have more options if you are open to going out in late May/early June. Sometimes, at least here in Summit County, there are more long-term rental options available at that time vs the Fall due to renters whom came in for the winter season, signed a long-term lease and then broke their lease once the season was over (does not happen often but does here and there.) those owners are left looking for tenants and typically there is not much of an influx of renters in late spring/1st part of summer.
Due to the fact you might have no car, again, I would keep tabs with property management companies, craigslist.com, online newspaper classified section (I assume Telluride has one) as it would seem the location is perhaps more important if you can come into Telluride at any point leading up to school starting.
If you are not seeing much online, you may want to post on Craigslist putting what you are looking for (it's free). Most rentals should have parking (might be surface parking without a garage) but I would be surprised for a renal to have no parking availability.
Car- If you can swing one, it's not bad to have unless you really are not going anywhere. Telluride is a cool place, but it is isolated so it might be nice to have a car to just get out of dodge once in awhile or for a long school break but again, that is up to you.
I would recommend not waiting till last second especially if you have school age kid as you might have to accept something you are not excited about. Start your search now via prop. management companies, online newspaper, craigslist as mentioned and be open to go when you find the right place (price and location). As the ski season gets closer, the more people start coming into town and then it becomes more competitive (prices might go up, depending on the telluride rental market...the whole supply/demand thing). Less demand in my opinion typically after ski season and that demand goes up as you get closer to ski season.
Best of luck.