Right now I don't think there is a way to add screenshots to the live chat because the Dynatrace support team should be able to access your Dynatrace environment via the SaaS login or the Management Console if you grant them access to your Managed instance. This would allow you to work dynamically with the team in order to find solutions much faster than screenshots should be able to manage.
Does that resolve your issue or there a special use case where screenshots would be more valuable?
If so, then you can submit a request for enhancement (RFE) to the forums for them to consider adding this feature.
Well said David. Thanks for helping out with this.
@Mitchell: The good news is that we will soon release a much improved live chat + messager feature that will allow you to paste in screenshots directly in your chat or attach files.