Praise be to Allaah.
Yes, you can be a Muslim without eating these animal products, but you need to be aware of the following:
O you who believe! Make not unlawful the tayyibaat (all that is good as regards foods, things, deeds, beliefs, persons, etc.) which Allaah has made lawful to you, and transgress not. Verily, Allaah does not like the transgressors. [al-Maaidah 5:87]
Say: who has forbidden the adornment with clothes given by Allaah, which He has produced for His slaves, and al-tayyibaat [all kinds of lawful things] of food? Say: they are, in the life of this world, for those who believe, (and) exclusively for them (believers) on the Day of Resurrection (the disbelievers will not share them). Thus We explain the aayat (Islamic laws) in detail for people who have knowledge. [al-Araaf 7:32]
Say: Tell me, what provision has Allaah sent down to you! And you have made of it lawful and unlawful. Say: Has Allaah permitted you (to do so), or do you invent a lie against Allaah? [Yoonus 10:59]
You should not think that it is better to abstain from these foods, or that doing so will be rewarded, or that a vegetarian is closer to Allaah than others, and so on. It is not permitted to draw closer to Allaah in this way. The Prophet (peace and blessings of Allaah be upon him), who is the best of mankind and the closest to Allaah, used to eat meat and drink milk and honey. When one of his Companions wanted to give up meat, he told him that this was wrong. Anas reported that there was a group of the Companions of the Prophet (peace and blessings of Allaah be upon him), one of whom said, I will never marry women; another said, I will not eat meat; another said, I will not sleep on a bed; and another said, I will fast and never break my fast. News of this reached the Messenger of Allaah (peace and blessings of Allaah be upon him). He praised and thanked Allaah, then said: What is wrong with the people who are saying such and such? I pray and I sleep; I fast and I break my fast; and I marry women. Whoever deviates from my Sunnah (way) does not belong to me.(Reported by al-Nisaai; the story is also to be found in the two Saheehs of al-Bukhaari and Muslim).
There is a great difference between not eating a certain kind of food because one does not like it, or it does not agree with one, or one has been put off for example by seeing an animal slaughtered when one was a child, which may leave one with a distaste for meat and other similar reasons, and thinking that meat is haraam and that abstaining from it is an act of worship, as is done by Brahmins, monks and others who are misguided.
Once this matter is clear in your mind, there is nothing wrong with not eating foods that you do not like. We will be happy to welcome you soon as our sister in Islam. We ask Allaah to give you the strength to do good and to protect you from every evil. Allaah is the One Who guides to the Straight Path.