Sorry it's actually easier than that. Here is what I just tested and it works.
Save the QIF file to your computer, let's say your computer is called Peter and the file is at c:\Temp on Peter.
In Reckon Hosted, go to File - Utilities - Import - Bank File
Browse in the top box and when you have the list of drives and folders, navigate down to PETER Local C Drive, then navigate to Temp and you will see the bank file.
Click to select the bank file, you will then be returned to Reckon. Next it will ask you where to save the file.
In this section select the Q Drive, which will be named per your User ID.
Once on your Q Drive, if you want to, you can create a new Folder and call it bank files for example.
Select anywhere on your Q drive (if you don't have a specific folder) and click OK.
Next (refer my image) your import should look something like the attached image.
Next select the bank account and OK.
The security issue from memory is when you login for the first time a permission box appears asking if you give permission to talk to your computer, however it seems you have said no.
I'm not sure how you can get that message back to say yes!
You could take a copy of your QIF file, then when you get to the import stage as Nathan mentioned and browse to a folder for bank files on your Q drive, if you click in the empty space you can press CTRL + V and it will paste.
However you can't cut & paste with Premium - you will need to temporarily login as Standard. Which is when you may also get the popup message about permissions?
In any case, for now, let's try the copy and paste method. You will need to be logged in as Standard. Then open Windows Explorer on your local PC so you can see your QIF File.
Open up Reckon Hosted as Standard, then go and open up your File - Import - Banking File process and hit browse on the top button to open a browse box and instead of browsing to your C drive, browse to your Q Drive - but don't go into any other folders.
Use the small New Folder icon shown in the below diagram:
Create a new folder called Banking Files, once created, click to go into that folder.
Now, ALT-TAB or switch over to your local PC where you have Windows Explorer open and right click on your QIF File and choose copy.
ALT-TAB or switch back to Hosted where it should still be sitting at a blank Q:\Banking Folders drive... now click in the folder then use CTRL-V to paste the QIF file in there. You can't right click so you have to use CTRL-V.
The QIF File should now be visible and selectable from your Q Drive.
Now in the Saved Converted File As box, browse also to Q Drive\ Banking Files, select your bank account and you should be away.
Copy/Paste does work in Premium but it's somewhat trickier and as you have access issues, please try the standard Copy Paste method above and see how you go.
Then maybe support can have a look at why you can't access your local C drive, as that would be much easier in all honesty.