Choosing a ML algorithm: is MLP + SHAP suitable for binary classification with small amount of data points but large amount of features? My data frame is around 32 data points (patients) x 1000 features (numerical blood biomarkers, eg. glucose concentration, protein concentration, etc.). I am classifying medical information with 1 or 0: diagnosis or no diagnosis (already given - so it's a supervised learning task). I want to show which of the feature or combinations of features best predicts the diagnosis.
I think using a Multilayer Perceptron (MLP) network to do the classification, with SHAP (Shapley Additive Predictions) to show the best predictive feature(s) may be the best bet.  Would anyone more knowledgeable recommend a different technique to start with? Is this even possible with sample size of 32?

32 data points are far too little to use any neural network, and it's very hard to extract any reliable information from such a tiny dataset with any method. Anything that comes out of your analysis is very likely simply noise. Sorry.
That said, your best bet will likely be a classical logistic regression with regularization. Take a look at GLMNet implementations, available in the glmnet package in R.
