All snakes are pure carnivores, though there are many that do not eat rodents, though not many are commercially available. All the larger snakes, pythons and boas, are going to need rodents or rabbits as a diet, though most will eat poultry as well. Smaller snakes, such as the green snake eat insects. Brown snakes eat earthworms. Egg-eating snakes, obviously, eat cheeseburgers... err, no, that's not right, they eat eggs. Many species of ribbon snake and garter snake can do just fine on minnows and small feeder fish. Vine snakes and flying snakes eat primarily lizards. There are even some less commonly available snakes, like the snail eating snake.
As for monitor lizards, you're talking about some 70 different species, at least a dozen of which are readily available in the pet trade. Some get big, some don't. Some will do just fine their whole lives on a supplemented insect diet, like Timor Monitors and Savannah Monitors, but obviously the larger the lizard, the larger the food item needs to be to get it enough nutrition. They can eat pretty much any meat, and they don't tend to be particularly picky. In the wild they're generally scavengers before they are hunters. The problem is, that whole-prey items give a better nutritional balance than feeding just random meat pieces. Organ meat, such as from the liver, provides essential nutrients that they can't get by just eating a drumstick. So for whole body nutrition, you really don't want to be skimping. Plus, poultry has a higher risk of salmonella, which is already a concern when dealing with monitor lizards. Once you start talking about Nile monitors or water monitors, they're big, exceptionally messy, and even a nice one can be a real chore to handle without getting yourself hurt.
Tegus tend to be omnivorous, and will eat a wide variety of insects, eggs, rodents, as well as fruits and some veggies. The ones I've had come through here don't seem to be terribly fussy eaters. They do need protein, but you could probably get by without feeding whole rodents, as long as you provided other protein sources, and supplemented with a vitamin powder of some sort.
There are many species of lizards that can be kept just fine on a purely vegetarian diet, iguanas are probably the most obvious herbivore lizard - but probably the worst choice for a pet. Some people get lucky with an iguana, but most of them end up in reptile rescues, or worse off. They are, by far, the most dumped reptile pet. Most other vegetarian lizards tend to be smaller than iguanas. Bearded dragons are probably what I consider the best choice for a pet lizard, and while they are omnivorous, and will eat all manner of insects - and even small rodents, they don't have to be a significant part of their diet for them to do just fine. Then there's various species of agamid and gecko that can be fed on a vegetarian diet, but they're definitely not on the larger side.
So what is your specific qualm with feeding rodents? If it is the live feeding thing, that is an easy fix. I buy all my mice and rats frozen and pre-packaged, vacuum sealed on Styrofoam meat trays. I thaw out what I need, as I need it, and feed it to my animals. My rabbits come from a commercial meat supplier. No rodent-eating snake or lizard requires live prey in captivity, and in fact it is dangerous.
I generally try to discourage people from keeping animals they might have qualms about feeding. If a natural part of its diet is just not something you want to deal with, a monitor lizard or snake probably isn't the best choice for a pet. They aren't a great choice for a pet for many people, as is. Especially the larger species.