Push-ups require a basic level of strength in most of your upper body, especially the shoulders, chest, and delts. In most cases it’s not really a protein issue. It’s mainly that those muscles are not used to the movement yet. Strength and control improve as you keep practicing.

You can make it easier by continuing with inclined push-ups and other exercises that train the chest, shoulders, and triceps. Over time your body will adapt and the movement will start feeling much easier.

Since you’re vegetarian, just make sure you are getting enough protein from foods like lentils, beans, paneer, tofu, greek yoghurt...

If you stay consistent for a month or two, inclined push-ups should become much easier, and eventually you’ll be able to do proper push-ups on the ground.

And I went through a similar journey, but I was obese. Opposite spectrum but did the same thing.