Is it permissible for me to convert to Islam while being a vegetarian (I do not eat meat or animal products)?

Yes, you can be a Muslim without consuming animal products, but you must be wary of two things: First: believing that these things are forbidden, for Allah Almighty says: (O you who have believed, do not prohibit the good things which Allah has made lawful for you, and do not transgress. Indeed, Allah does not like transgressors). Second: believing that refraining from eating animal products is better or that one will be rewarded for it, because the Prophet, peace and blessings be upon him, ate meat and drank milk and honey. He disapproved of those who wished to worship by refraining from meat, for in the Hadith (definition): "What is wrong with some people who say such and such? But I pray and I sleep, and I fast and I break my fast, and I marry women. So whoever turns away from my Sunnah (definition) is not of me." There is a difference between one who abstains from food due to lack of desire or natural aversion, and one who believes it is forbidden or worships by abstaining from it.