One of possible approaches may be to perform so called optimal scaling on your 4 ordinal response variables. This a procedure to transform ordinal scale into interval scale under the aim to maximize linear correlation among the dependent variables or between the dependent variables and some independent ones. (Some techniques using optimal scaling.)
I suppose you are interested in the latter aim and so you will need Categorical canonical correlation analysis by optimal scaling (OVERALS). Like linear canonical correlation analysis, it maximizes correlation between sets of variables, in your case - the dependent set and the independent set, - but it does it allowing for nonlinear transform of "ordinal" variables into "interval" ones. Aptly may be noticed here that usual (linear) MANOVA is (closely related to) linear canonical correlation analysis, so saying "I want MANOVA" is echoed by "take canonical analysis".
The optimal scaling method just described is an alternative to multilevel generalized linear modelling suggested by @Jeremy in his comment. I can't say which way is "better". The difference between the optimal scaling and the generalizing approaches is that the first assumes a specific mode of explicit transform from ordinal to interval "underlying variable", whereas the second assumes a specific kind of distribution for that interval "underlying variable" by utilizing a corresponding implicit link function and minimizing errors w.r.t. to the observed variable.