Can I use logistic regression when all of the regressors sum to 1? Let's say I want to perform a logistic regression (binomial) as :
X ~ P1 + P2 + P3 + P4 + P5

where X is binary variable (0 or 1) and P1, P2, P3, P4, P5 are proportion of cell type ( e.g. P1 = proportion of T-cells ; P2 = proportion of B-Cells, etc..). The sum of P1, P2, P3, P4 and P5 is 1 for each observation.
Can I perform a univariate regression for each Px variable e.g X ~ P1 and a multivariate regression with all Px variables knowing that the sum of all Px is always 1 ?

Variables representing parts of a total, like your proportion of cell type, are called compositional-data, for more information you can peruse that tag.
But in your case the solution is simple. Your design matrix will be rank-deficient, since the sum P1 + P2 + P3 + P4 + P5 will equal the intercept. That way the intercept is included in the design matrix in two different ways. The most practical (that is, interpretable) solution is to omit the intercept from the model. In R that would be something like
glm(X ~ 0 + P1 + P2 + P3 + P4 + P5, family=binomial, 
    data=your_data_frame)

The ~ 0 + part of the formula means omit the intercept.
User whuber points to another sort of analysis, using logratio transformation. See How to perform isometric log-ratio transformation
