It's a private building and MLSE can refuse entry to who they want. Same with every other private establishment.
I am not interested in an argument of whether you should/could/would get vaccinated. However, in a case of entry, business can 100% outline the need for a vaccination card or vaccination exemption card to allow service. Just like, unless mandated by the province, there will be businesses who don't require it. Just like how you can decide to get vaccinated or not, private businesses and locations can decide, outside of religious, gender, race and sexual orientation discrimination, who they serve, especially since it would be considered a employee and public health decision.
So quite frankly, take away any discussion of getting, or not getting vaccinated, trying to sue MLSE (lol) for not permitting you entry into their private location, wouldn't go well, and would be quite the waste of your time and money.
I'm of the demographic that probably won't be eligible to get the vaccine until, assumingly, next summer or the fall. I am assuming, the same thing can be said about you. That means vaccines will have gone out WORLD WIDE, to the most high risk population. So by the time we are allowed to get it, there will be more than enough evidence that these vaccines are no less unsafe than a normal flu vaccine.
For the record, I don't get the flu shot yearly. Only year I did, was the year of the swine flu surge, and funny enough, I got the flu that winter. It's pure coincidence. However, I am naturally someone that constantly washes my hands, and uses sanitizer, whenever I deal with people. I don't tell other people not to get it, and I don't get angry at people who do it. I don't tell people to not vaccinate their kids. I simply make the choice to not the flu shot myself, in large part because of the coincidence, and it's just that, as well as the general precautions I take otherwise during flu season.
Vaccine reactions are fairly common. Generally, the arm you get it in will feel weaker for a couple days, you might feel a bit more tired, could have a headache, and so on. It's all normal. With any vaccine, those who, generally, have an egg allergy, are at risk to have an allergic reaction. Again, it's uncommon for the general pop. however not that uncommon for the population with egg related allergies.
(and for anyone other than OP reading this - yes, I do plan on getting the COVID vaccine when I am eligible - because it's important for everyone to do

)