Thank you fro getting in touch and providing the credentials as well.
In order to rule out a plugin conflict being the possible cause of the issue I would recommend temporarily deactivating all your non-toolset plugins and checking again to see if the problem still presists.
I took a look at this for you and copied the site to my localhost.
When I checked locally i am able to import the Types components without any issues at all. It would appear that what is happening is server based and given that there are no errors on the page i would suspect a missing module for importing of zip files.
However checking further I see that you are able to import Forms components which indicates that your case is quite unique. There is an alternative in that you can use our Module Manager plugin to Import/Export Toolset components.
The module manager plugin can be used to Import/Export Toolset components. If you go to Toolset -> Modules you can export various Toolset components as modules rather than exporting all your CPT from Types you can select which ones you want to export etc.
Then to import these modules you will go to Toolset -> Export/Import
If there are further questions please let me know.
