Upgrade Peugeot 5-speed to indexed flat bar shifters
If the goal is to make it easy and work well, for sure get a modern freewheel, like a Shimano 6-speed hyperglide one. Hyperglide style ramping is the glue that holds indexed shifting together. Even if you jumped through all the hoops needed to try to make it index with the freewheel you've got, it wouldn't work.

A lot of gripshifts threaten to take up too much room on the sort of bar it probably has, so good idea to measure that ahead of time (adding in the length of the grip you'll use).

If it can fit them, then a good cheap approach would be getting 6-speed MRX comps with the semi-index aka non-index left (lots of small clicks) plus almost any cheap MTB/hybridy rear derailleur and the aforementioned freewheel, plus a new chain. You could even get the rear gearing lower this way, which is good because that sort of bike didn't come with very thoughtful gears. You could do the same thing with similar thumbshifter sets, where the left is friction and the right is 6.

Making the front index is a whole other ballgame and usually would require replacing the crank before it works smoothly, which isn't necessarily bad depending on how hard you want to commit to the bike.