My husband and I spend one night and two days in Odawara and I thought the following tips might be useful for vegetarians:
If you leave the Odawara train station via the western exit, you'll see the Hakone Bakery, which consists of a take-away counter and also a restaurant and cafe. In the morning there is a vast array of freshly baked items including a delicious red bean paste pie, sandwiches, buns, etc., and in the cafe there is a wood fired pizza oven.
I must recommend a little Indian restaurant which we found within walking distance of the Railway Station. I don't have its address, but it's called Shakti, and its located in the basement of the Seiyu supermarket. Just write S-E-I-Y-U on a slip of paper and ask locals for directions to Seiyu, which is a really big supermarket. I doubt whether too many locals would know where the Shakti is, but they would know the supermarket. Shakti restaurant has a delicious, small range of vegetable and dahl dishes as well as non-vegetarian, and they baked the naan for us while we were waiting. It's very inexpensive, and a terrific experience. It's located in a narrow street with no cars - only pedestrians, with small shops and stalls. I hope you find it.