In this desmos graph I'm trying to understand epicycles and want to know if there is a way to leave the trace for the locus of the green and red points:
By default desmos is creating a slider for t instead of a continuous range [0, 2pi].
But is there a way to tell desmos to create a range and not a slider?
Yes. There is a way to make ranges, using the special variable $t$. Since you used $t$ as a slider in your graph, I had to modify it in order for the ranges to work. As @soupless said, if you take a look at Paremetric: Cycloid, it shows how you can follow the path of any functionally defined curve. In your case, because you use the function $v$ in more than one line, I would set $t$ as a parameter instead of a global variable. That way, you get a more compact form as shown in this graph:
However, since you want the locus, not the path traced by each point, just replace each $t_1$ with $2\pi$:
Now you can find the locus of any epicycle, not just the one you described.