If you mean to go in the near future your stroller would need to have Snow Mobile Skids on it - It's slushy and getting more snow as we speak.
If you mean in the dry months it's possible - but I would make sure it's legal for one thing - some areas do not allow mechanical devices of any type - state park rules apply from the tram to the other side of Wellmans Divide and then you enter National Forest, check with both agencies to make sure.
As Hiking Jim pointed out, there are some areas you will be carrying the child and some areas that you could push the child.
I've seen quite a few folks carrying their kids in the adaptive child carrying backpacks - One parent carries the child and the other carries the water and other hiking 10 essentials.
The distance from the tram to Idyllwild is only 8+ miles with less than 1,000 feet of elevation gain over Wellmans Divide then mostly downhill to Strawberry Junction and then really downhill on Devils Slide.
You might do the trail once without the child to see if you think it's doable then attempt it with the child if you think you can navigate the tougher spots.
Good Luck
Lilbitmo