You can review the Blizzard EULA for yourself here.
The relevant bits, highlighted for emphasis:
“You may install applicable components or features of the Platform (including the Games) on one or more computers or mobile devices under your legitimate control.”
" License Limitations. Blizzard may suspend or revoke your license to use the Platform, or parts, components and/or single features thereof, if you violate, or assist others in violating, the license limitations set forth below. You agree that you will not, in whole or in part or under any circumstances, do the following:
i -
ii -
iii -
iv -
v - Cloud Computing: Use the Platform, including a Game, in connection with any unauthorized third-party “cloud computing” services, “cloud gaming” services, or any software or service designed to enable the unauthorized streaming or transmission of Game content from a third-party server to any device."
And for clarity, there are no authorized third-party cloud-computing or -gaming services.