If not, is there any extension that allows this? Also if not is there any 'workaround' for this?
Magento includes an out-of-the-box integration with UPS.
This is a link to the Magento 1 configuration (I didn't found the M2 example) but is should be really similar: http://docs.magento.com/m1/ce/user_guide/shipping/ups.html
Thanks for the reply, Damian. However, this does not answer my question. I understand that Magento integrates with UPS by default as do most shopping carts. However, what I need is not to have customers use our rates, but rather enter their UPS account at checkout, which would essentially be "free" shipping from us and they would be charged on their UPS account. Many of our customers are businesses, so half of them use their own account on our current website.
I wasn't sure enough but now I understand.
That is the out-of-the-box integration. You can explore this options too: https://marketplace.magento.com/catalogsearch/result/?q=ups#q=ups&idx=m2_cloud_prod_default_products...
(But I'm not sure if you'll find the option you are looking for)