Yes, you can be a Muslim without eating these animal products, but you need to be aware of the following: You should not think that these things are haram (forbidden), because Allah says (interpretation of the meanings):
“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, Allah does not like the transgressors.” [al-Maaidah 5:87]
“Say: who has forbidden the adornment with clothes given by Allah, 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 Allah sent down to you! And you have made of it lawful and unlawful. Say: Has Allah permitted you (to do so), or do you invent a lie against Allah?"