This can get complicated as it depends on how strict you are.
Many vendors use noodles that are at least vegan-safe, but of course, the condiments and other ingredients aren't.
At Hawker Centers and Kopi-tiams, and even some food courts, there are "Chinese Vegetarian Stalls." These are similar to the Bee Hoon stalls I mentioned. Your request for noodles and select the accompanying vegetarian dishes. There are sometimes also other options such as vegetarian wanton noodles, vegetarian stir fry crispy noodles, etc. These stores will usually prominently display the Chinese character 斋 i.e. vegetarian.
Outside of these, there are Chinese and Indian vegetarian restaurants too. Such as Lingzhi Vegetarian and Annalakshimi.
Finally, some restaurants offer vegetarian selections. However, some stricter vegetarians might not be comfortable with these because of the utensils, etc.