You should have no problem handling multiple requests simultaneously. How do you run your app? If you use built-in server (perl your_app.pl) then by default it is single threaded and will only process one request at a time.
You might want to use mutliprocess/multithread deployment options, for example Starman. It is described in https://metacpan.org/module/YANICK/Dancer-1.3113/lib/Dancer/Deployment.pod#Running-on-Perl-webservers-with-plackup