Hi Jerry. Thank you for the question.
There are no companies I am aware of who specialize in QuickBooks integrations. QuickBooks (Intuit) provides a framework for developers to leverage in order to create web sites to integrate with QuickBooks.
This is a link to the Intuit Partner Platform (their framework for developers to integrate)
The best thing to do would be to find a local web developer who is capable of using an API he/she has not used before. Likely they haven't worked with QuickBooks API before, but if they are a good developer they should be able to pick it up easily.
Please let me know if you have any questions and I will be happy to assist further.
With the # XXXXX businesses that use qbooks, you would think their are firms that specialize in integrating web sites to Quickbooks.
Where would you look for these firms?
These links are a good start!