Wish that I could reach right out and touch it.
When you see this little fellow, you'll want to reach right out and hug him to death.
Right in the cited example is really just an intensifier. For a similar usage, consider...
1: We left after paying the bill
2: We left right after paying the bill
3: We left straight after paying the bill
4: We left immediately after paying the bill
Where the right, straight, immediately versions all emphasize how short the delay was between paying and leaving. In OP's example it's not so clear-cut exactly what is being emphasized / exaggerated, but a couple of reasonable paraphrasings are...
5: I wish I really could reach out and touch it
6: I wish I could stretch out and touch it
...but it's also quite reasonable to say to reach right out implies to stretch out to reach.
You may have noticed that although I used really as an intensifier in my own example #5 above, that same word is effectively being used with the opposite sense in the first sentence of my answer here (as is the word effectively in this very sentence). If you did notice that, and/or find this slightly confusing, you might like to have a look at Opposite of “intensifier”? as asked on SO Linguistics some years ago. Some words can either add or reduce emphasis, depending on the context in which they're used.
My point is that it's not always worth trying to assign specific meaning to words like right, really, fairly - sometimes all they're doing is adding or toning down the overall "force" of some assertion.