Here are some easy steps to fix the missing button.
Step 1: Go to the three dots on the right of Chrome and go down to Help>About Google Chrome.
Step 2: It it needs to update it will do so automatically or if it is up to date move to step 3. After it updates, it should ask you to relaunch, relaunch it.
Step 3: Clear your Browsing Data by following the instructions linked Here.
Step 4: Go to the three dots on the right of Chrome and go down to Extensions>Manage Extensions
Step 4: Find the Proctorio extension and click Remove.
Final Step: Now close Chrome completely. Reopen Chrome and attempt to join the exam again. It should prompt you to install Proctorio. Follow the instructions and reinstall, and then your Take Quiz button should reappear.