I have data that I apply a test called Firth's Bias-Reduced Logistic Regression.
I am using R with package called brglm. I got coefficient and I will try to find a way to calculate a confidence interval
Is it valid to take exponentiation to calculate and report odds-ratio in this type of test?
In short, yes.
If you have coefficients on the log-odds scale, which is what Firth's penalized likelihood (or bias-reduced) logistic regression reports, using exp(coefficient) gets you an odds ratio. In fact, that is more than anything a core motivation beyond the particular method: getting estimates of coefficients/odds ratios that remove the first order bias term. It so happens that this has some nice side effects such as dealing with complete separation in a way that may be considered reasonable (but not reasonable if you think that some things might be completely causal to yes or no).
Confidence intervals work the same way: get the confidence intervals (or if you have only standard errors use $\hat{\beta} \pm \text{factor such as } 1.96 \times \text{SE}$) and exponentiate the limits.
