If religious exemptions don’t count when it comes to LGBT discrimination, where the basis is it harms other people, it should be the same logic for vaccines when necessary, as they harm other people.
It’s a pretty consistent idea that having a religion should not give you an excuse to do something that harms other people without consequence.
What are you referring to?
There are plenty of states that have laws that don't protect LGBT people from (negative) discrimination in some environments, including medical environments. If you mean a recent SCOTUS decision, that was worded in a way to include LGBT people in with discrimination through, which is illegal on the federal level (e.g. you can't punish this man for dating a man when you are okay a woman dating a man). In any case, the reasoning for LGBT protections isn't relevant here since LGBT protections are based on negative discrimination on a protected class.
And while some states agree with your reasoning, most have laws protecting religious groups and even moral reasons to not be vaccinated. Furthermore, it doesn't look like the exemptions are a major reason for high or low vaccination rates among children, so attacking the exemption seems pointless at actually getting a relevant number of children to be vaccinated.
More generally, while states can establish consequences for not being vaccinated when trying to make a mandate (e.g. fines), they can't force anyone to be vaccinated even with a mandate. Even mandating vaccination, though, will fall under scrutiny and be challenged in the courts because they can really only impose a mandates under specific circumstances, the last of which being, if I remember correctly, when there was a disease with an 80% fatality rate decades ago that could possibly be spread around.
