« Je le veux » is actually okay to my ears, though not idiomatic according to the native speakers who have weighed in. Arguably this le wouldn't replace the infinitive but whatever substantive is implied by the infinitive (e.g. « un voyage »).
A more common solution is to replace the verb by another. The general one is (le) faire :
Pourquoi veux-tu voyager ?
Je veux le faire parce que je veux visiter de nouveaux pays.
It might seem like « le faire » hardly saves time and effort over repeating « voyager », but two points may be made: that this isn't essential for pronouns, as long as they help you avoid repetition; and that the mileage will vary depending on the length of the original phrase, since it can replace everything branched under the verb:
Si, elle t'a dit qu'elle ne consentira plus de te voir autant qu'elle vivra, mais elle l'a fait dans le seul but de te rendre jaloux.
You could also say l'a dit here, repeating the verb but not the rest.
Also see Gilles' comment about possibly using a different verb depending on the situation. For example, if you want to voyager to a specific place, you might replace that verb phrase by y aller instead of le faire. This makes sense to me. I wonder if there might be a few atomic verbs that "catch" different categories, like faire, aller, être, avoir (these last two certainly can't be replaced by le faire).
Finally, Dimitris' suggestion is certainly applicable to everyday conversations if you just meant that you were looking for a practical way to avoid repeating a verb.