First up if you're a coffee drinker the Station Coffee house downtown Medicine Hat is fantastic and they have the best chai tea soy latte I've ever tasted though no vegan food options. You can also wander the rather quiet but interesting downtown where free parking is plentiful and crowds are absent. Across the river you can stop in at the best eatery in town, the Zucchini Blossom where they have clearly marked vegetarian and vegan menu items. I found they were happy to accommodate substitutions to make the vegetarian items vegan. I'd recommend their chilli, sandwiches/wraps and if you are so inclined they have a really good vegan cookie. The restaurant is clean, friendly and a hidden gem in the city that most of my family had never tried until I dragged them there.
Another great option is Fresh Healthy Cafe on the south end of the city. This place has a fast food chain kind of vibe but there are numerous vegan options including a vegetarian wrap which can be made vegan by subtracting the feta cheese, smoothies (and smoothie bowls) and various other salads and wraps that they will substitute vegan mock meats for the chicken that is otherwise in the food. Cocoloco is a unique shop tucked away in the same building as the Greyhound Bus station downtown. They sell coconut oil, purporting it to be a health food (it's not), mushrooms and cacao products including some tasty locally made chocolates. The owner is enthusiastic and will talk at length about health and spirituality so that may be interesting (or not) to some people.
Other chain restaurants like Original Joe's, Subway and Mucho Burrito have vegan options, but my thought is that when you are visiting a place there is no point in eating food you can easily get at home. The best grocery stores are the Real Canadian Superstore (which is one of the only places you'll find selections of tofu, tempeh etc) and the Co-op grocery store with the best produce. They do have farmers markets where you can get local produce and greenhouses where you can buy produce year round. If you happen to be there during the Taber Corn season (August) be sure to stop at one of the stands for the sweetest corn on the cob you'll ever have. Saskatoon berries are freshly available in late June/early July and are a nice treat for a unique berry you may not get at home.
Overall, the vegetarian culture isn't really present in smaller prairie cities and towns in this area. The culture is dominated by beef (cattle farming), lake fish (like trout and pike), and plenty of eggs/dairy. I've yet to come across another plant based person on my frequent visits and the food choices I make are normally reacted to with bewilderment. Of course there must be some vegetarians there and they make do with their environment and thrive. Once you figure out your food, the city has plenty to offer including a massive number of well taken care of cycling and running trails, great recreational facilities and a low crime rate.