Yes, this is possible, and you should do this when you are screening for some variable that predicts (or correlates with) the log odds. This would also be required if the screening is done in a multivariate model.

But be aware that the multivariate model and the univariate models do not test the same hypotheses. Make sure you the hyptheses that are meaningful in your context, and not hypotheses that are not particularily meaningful but easily testable.

Just some thoughts....

So, you'd run 10 models, and then retain (?) only those independent variables that have a p-value less than 0.005 ? Is this what you want ?

Is there a better way to choose a model with a subset of those independent variables ?

Or maybe is it more meaningful to report the univariate p-values (with some kind of effect size measurement) without judging if they are "significant" ?

Another thought: if you are interested in the correlation between an ordinal variable and a continuous variable, maybe use correlation for each pair ? (Maybe Spearman or Kendall tau-b ?)

Plots for each pair of ordinal and continuous variables would probably be helpful to look at. You may need to "jitter" the points on the ordinal variable so you can see all the data points.

Two points before I can offer an answer:

1. Are you interested in how each of the 10 are associated with the response variable, or are you interest in how well some subset of the 10 can be combined to predict the response variable? Or do you have more specific research questions like does variable A predict Y after conditioning on variable B.

2. Say more about why including all ten does not work. Is the sample small? Are the predictor variables collinear? etc.?