I have the following dataset shown below. Any value between 500 & 900 were categorized as A, while values between 900 & ~1500 were mixed between A and B. I want to find the probability of getting A, B, and C at any value of x where x is my independent variable and A,B,C are my dependent variables. It seems to be a good fit for multinomial logistic regression. I believe the number of observations for each dependent variable is sufficient. If multinomial log regression is appropriate, I wish to uses Python's scikit learn logistic regression module to obtain my probability of A, B, and C at any value of x but I am not sure how to approach this using that module.
Your outcome is ordered, but a multinomial logit would ignore that, since it is a model for a categorical dependent variable with outcomes that have no natural ordering (like bus, train, car for mode of transportation).
I would consider taking a look at ordered logit/probit since that seems more like your setting and can produce probabilities for each class.
If you want to predict a single class, or if you are working under the assumption that each sample, $x$ should have one class then, yes Multinomial logistic regression is the right choice.
If on the other hand, you want to allow the model to assign more than one class for an example maybe $x$ could be assigned both $A$ and $B$ then you may want to use multiple (3 in your case) independent binary logistic regression classifiers.