You didn't say how many speeds, which is the most important parameter here. From what I googled about Trek 720 multi-track and Gripshift MRX 100, it seems your bike has a 7-speed cassette or freewheel.
7-speed cassette ideally should use 7-speed bar-end shifter, or secondarily if that's unavailable 8-speed bar-end shifter with one unused click (8 speed systems use very similar spacing so they are reasonably cross-compatible). Note 9 speeds with two unused clicks don't work due to the different spacing. They don't make these new anymore. You may or may not be able to find such shifters on eBay, either used or new-old-stock. If there was a 8-speed new-old-stock bar-end shifter available, I would have already bought it to expand my largish collection of 8-speed new-old-stock bar-end shifters, so that's a likely "no". But you may be able to find used 8-speed bar-end shifters or even 7-speed bar-end shifters.
You also didn't say what brakes you have. A quick googling suggests cantilevers. If so, they use standard pull brakes so for example Shimano BL-R400 drop bar levers should work. Note that flat bar levers usually have barrel adjuster but drop bar levers don't so you should fit Shimano SM-CB90 adjuster for both front and rear brake housing.
Your bike may be a very good candidate for drop bar conversion. The only big questions are:
- Can you find bar-end shifters from eBay that match how many speeds you have?
- Is the frame size suitable for you with ideally the current stem or perhaps with a slightly shorter stem?
Today, such a conversion would be very difficult with "modern" bikes. The pull ratio of rear derailleurs is different between "road" and "MTB" parts. Also brakes are usually hydraulic, and they don't make hydraulic drop bar levers without integrated shifters -- and those shifters are available only for "road" rear derailleur cable pull ratio.