If you've done your research you will find two primary canoe trip outfitters in Whitehorse, although there may be others.
Kanoe People
http://www.kanoepeople.comand
Up North Adventures
http://upnorthadventures.comI have paddled Whitehorse to Dawson 5 times (thus far), including twice continuing beyond to the Dalton Bridge. But these were offical races, not with a tour company. Kanoe People has been very friendly to us with supplies and use of their riverside facility for storage, launch, and minor repairs to our racing canoes. I would recommend them, but Up North is also a good company from what I know.
I have only paddled the river in late June on the Yukon River Quest race to Dawson, or in July on the Yukon 1000 mile race to the Dalton highway bridge. June typically has higher water levels (making the 5 fingers rapids a real fun rush, but through it entirely too quickly - don't blink or you'll miss it). Water levels can change significantly from one year to the next. July will offer lower water levels and less rapid current overall with more shallow water gravel shoals with fewer opportunities for short cuts through sloughs or bypasses around islands. Efficient navigation is more complicated then. Later in the season would likely have even more shallow water and exposed gravel.
Why not take in the whole experience and do Lake Laberge as well? It is 33 miles long, and the river from Whitehorse to the lake is spectacular in its own right. We always take a pre-race day training run or two from WH to Takhini, or just before the beginning of the lake. Multiple times we have lost count of the number of bald eagles (more than 30) seen along the river in that section of river, many more seen there than in any other section of the river. The view of the approach into the lake is tremendous and is always one of my most memorable and awesome experiences. Although we did not need them, there are many potential landing and camping sites along the eastern lake shore. The whole region of high rocky cliffs and ridges seems to call out, saying "hike me".
The only negatives to paddling the lake are, well, it is 33 miles long. Plan for a couple of days at tourist speed. And the Lake creates its own weather. I have paddled it in oppressively hot temperatures in still air with glass-like water surface and no wind whatsoever. Other times with 3 changes of wind direction and stormy weather systems with crashing white topped waves during the 5 hours it takes a canoe in race mode to traverse its length. But hey, it is the famous and historic Lake Laberge! Not to be missed IMO. Who knows, you just might see Sam shivering somewhere along the marge.