The only natural products that give significant sun protection to the lips (SPF 15 or higher) are the ones containing zinc oxide. They do work quite well as a sunscreen, but the zinc in them makes the lips white (which most people see as a downside).
If you are willing to consider minor sun protection, just plain shea butter (the unrefined kind) offers some lip protection (around SPF 4). Which is better than nothing, but not really enough for a sunny vacation and spending lots of time in the sun. However, it is all natural.
All the non-whitening lip balms with sunscreen contain chemical sunscreens, which are generally dangerous. However, not using sunscreen on the sensitive skin of the lips is also dangerous.
All in all, the situation is a mess. I'm sorry there is not a better answer.
EDITED TO ADD:
Of all the zinc based sunscreens, this one has the best reviews as far as not turning the lips white:
There are still a few complaints about it being kinda sticky and possibly difficult to apply, but it is NATURAL, protects well (SPF 30), and does not turn your lips white (or at least much less white than the usual zinc products).
It seems worth a try, I just put a tube in my Amazon cart.