In R, two main functions ("logistf" and "brglm") can handle Firth's method. MuMIn allows to extract AIC(c) values for both functions, so my first guess would be "yes".

However, quoting from the "brglm" vignette:

The use of Akaike’s information criterion (AIC) for model selection when method = "brglm.fit" is controversial. AIC was developed under the assumptions that (i) estimation is by maximum likelihood and (ii) that estimation is carried out in a parametric family of distributions that contains the “true” model. At least the first assumption is not valid when using method = "brglm.fit". However, since the MLE is asymptotically unbiased, asymptotically the modified-scores approach is equivalent to maximum likelihood. A more appropriate information criterion seems to be Konishi’s generalized information criterion (see Konishi & Kitagawa, 1996, Sections 3.2 and 3.3), which will be implemented in a future version.

Furthermore, when trying to fit the same model using "logistf" and "brglm", parameter estimates are similar, but the AIC values are very different. Specifically, "brglm" returns a value of AIC similar to that of "glm", while "logistf" returns a much lower AIC value.

Any clue why that happens?