I would like to know if there is a way to build a pseudo panel dataset using repeated cross sectional survey data by matching individuals across rounds using something similar to propensity score matching, and then run a probit model.
Since you plan to 'run a probit model', you are probably thinking of some GLM panel data model that models individual-specific unobserved heterogeneity. But creating a panel when there is none is not advisable in my opinion. Even if you are able to match on the covariates or the propensity score with enough observations and use an accurate matching criterion, maybe even exact matching, there might still be unmeasured heterogeneity between individuals, which is what you could solve with 'actual panel data' in the first place. Doing that on a fake panel probably introduces large biases. After all, if you would be able to measure everything, there would be no need for a panel data model as unobserved heterogeneity will be accounted for.
If you still want to do it for some other reason, accurateness and as little omitted variables as possible is a necessity if you want to pretend to have a panel by using matching. Make sure you have a sufficient amount of observations for all cross section panels. Also you should have a enough variables that are predictive of the outcome as well as correlated with the time period, so as to estimate an accurate propensity score. There should also be sufficient overlap between time periods in terms of covariates, in order to match enough observations. It probably should be fine to work with only a subsample suitable for matching, which is probably the only feasible option, due to the accurate matching required, perhaps even for multiple time periods. In the process you probably forgo any representativeness of the sample for some population of interest.
After matching you could have your 'pseudo panel'. However, I would advise against it and just keep the data and analyse it as cross sectional, e.g. using difference-in-differences if you have multiple treatments. Even then, you can do propensity score matching or other treatment analysis methods over time or groups to address confounding due to the different samples.
Besides what Nick mentioned, I belive that using ps-matching for pseudo-panel is extremely dangerous since your dependent variable (Y) in your probit or logit model will take the value of "1" if the individual belongs to survey 1 and 0 if she corresponds to survey 0, consequently, you will be modeling the probability of belonging to one survey or another, which is equivalent to modeling pure randomness.