I'd be happy provide some insight on your issue.
In this situation, your best option is to use a third party app that directly integrates with QuickBooks Online. There isn't a way to transfer just transaction information from your Desktop file without transferring the entire company file. In addition, transferring the entire file is only available during the first 60 days of the Online account.
Here's the link to apps that are available to be integrated with QuickBooks Online: https://apps.intuit.com/ .
Please let me know if you have any other questions and I'd be happy to help. Thanks.
IF web connector transactions are in an excel sheet, or you can get them in an excel sheet
This utility will take that excel sheet and import to QBO
I am not affiliated
or, click the products menu at the top right, and they have a utiltiy to take transactions out of desktop, then you could feed them into the utiltiy I linked
While this answer will not help you in the short term, it may in the long term. I am in the process of developing exactly what you are asking for but it is still a ways out.
However, I am happy to spend some time showing you how we currently complete the process, it is not a very simple task but once you get it down, its not that bad.
I have played with most of the importer | Export programs on the market and find that I like CData or Zed Axis best for your situation. One license will work for both QuickBooks Online and QuickBooks Desktop.
CData is a little more challenging to use at first, but it is heavily excel based which has allowed us to create a couple VBA macros to help automate more of the process. Here is a link to a video we created demonstrating CData.
Since every persons process is unique, I would like to offer to take a couple moments with you on a phone call to listen to your objectives and then to share my experiences if that would help.
If intrested, feel free to lookup my profile and give me a call.