My girlfriend and I spent a week in Chiang Mai before heading up north to Pai. Our last stop for lunch was at Aum Vegetarian in Old Town. We had our giant backpacking packs with us. The staff were incredibly friendly and let us leave them near the host stand while we sat upstairs to eat. The upstairs dining area was full of comfy places to relax while waiting for our food. We ended up hanging out there for a couple hours after eating before heading out of town.
After returning from Pai a week later we decided to make it our first stop for lunch before going to our hotel. The staff immediately recognized us and were excited to hear our thoughts on our time in Pai.
The food was amazing both times that we ate there. I would recommend you give it a try while visiting Chiang Mai whether you are a vegetarian or not. 