On both Firefox and Google Chrome, you can use the scroll wheel to change volume on YouTube video. However, unlike Chrome, you will have to use add-on for Firefox to be able to do it.
For Chrome users, all you need to do is simply hovering the mouse cursor over the volume button on the video and scroll the mouse wheel up and down to raise or lower the sound level.
For Firefox users, while it is true that you have to use an add-on, this one allows you to change the volume by scrolling your mouse wheel anywhere on the video. In addition, you also have plenty of other features related to YouTube such as stopping video autoplay, grid layout in search results and subscription, blacklist unwanted channels, grab video thumbnail, take screenshot, and lots of others.
The Firefox add-on that can do all the cool stuff mentioned above is YouTube Plus.
The option to change volume using the mouse wheel is disabled by default. Here is how to enable it.
First, install the add-on from here.
Then open YouTube site. You should be able to see the new sound button at the top right corner. Click on it to open the settings page.
Now select the Video settings (1), put a check mark on the “Change volume with mouse wheel” (2), and hit the Save button (3) on the upper right side next to the Reset button.
Open a new YouTube video to see if it works.