Even in rather small places there are vegetarian restaurants in Taiwan. I was eating the other day at this place, next to the Visitor Center of Shueishe in Yuchih Township, where the bus stop for Sun Moon Lake is.
Even if you only speak like 5 words of Chinese as I do, you can order. At least one of their menu cards is in English. Grab it on the table and write down on a piece of paper what you want to eat.
I had a soup with rice and the small cabbage dish. The cabbage was very delicious. The soup had some tasty crunchy-like things in it and lot's of rice of course. The broth wasn't that tasty though - to add something salty helps.
I payed for each dish NT$ 100.
Sun Moon Lake is very touristy and therefore rather "expensive" compared to other places in Taiwan.
Since this restaurant doesn't close the door it can get kind of chilly on a January evening to sit there: bring your jacket and a sweather ;)
It seems to be ok to bring your own drink with you at cheap places when they don't sell beverages in Taiwan as far as I noticed.
Please note, that the 3rd shop on the left, counted from the visitor center (Wu Sho Si Hai is the second one), also has the chinese charakters 附設素食 written in green on the sign (the last two are for vegetarian food). Google translate shows "vegetarian attached" for this combination (please correct it, if it's wrong).
They have a couple of vegetarian dishes, but there is also meat served. There is an English menu available but it is not complete as I found out.
The nice thing was, at this place they can close the door.
At the moment in January it is like 17°C/63°F during the day, but at night it gets well below 10°C/50°F.
If you like a sweet fruits dessert, I can recommend to buy an Atemoya (don't confuse it with the sugar-apple!)
at one of the different fruit places at the same street (I payed NT$ 70 for one). If it's kind of soft, it's ready to eat. Just peel it with a knive and eventually cut it They might do it for you when you buy it. You also can cut it in half to eat the inside with a spoon.
My second favorite ist passion fruit (cut it in half and eat with a spoon) and ripe Guava.