Hello there,
Vegetarian is not popular in Vietnam, but don’t worry too much about this because you can find some vegetarian restaurants in big city. And in mountainous areas or countryside is not big problem. As you know, Vietnamese people often eat frugal food such as kind of vegetables or tofu, so ordering vegetarian food in these places is definitely possible. You can consider vegetarian restaurants in big cities as I recommend below:
In Hanoi:

1. Loving Hut at No. 4, Alley 192 Quan Thanh, Ba Dinh, Hanoi
2. Zenith Yoga studio II & Cafe at Alley 14 Duong Thanh St, Hoan Kiem, Hanoi
3. Nang Tam vegetarian restaurant at 79A Tran Hung Dao Street, Hoan Kiem, Hanoi
4. Swan vegetarian Restaurant at 14 Duong Thanh, Cua Dong, Hoan Kiem, Hanoi
5. Amitayus Vegetarian Restaurant at 184B, Xuan Dieu Street, Quang An Ward, Tay Ho District, Ha Noi

In Ho Chi Minh:

1. Hum Vegetarian, Lounge & Restaurant at No.2 Thi Sach, Ho Chi Minh, Ben Nghe
2. Phuong Mai Vegetarian Restaurant at NO.7 Phan Chu Trinh, Ben Thanh, 1, Ho Chi Minh
3. Buhha Chay at No. 31 Dang Tat, Tan Dinh, Ho Chi Minh

Hope this information is helpful for you.
Have a nice trip!