Help! Auto-focus not working on Manual Mode but working perfectly fine on Auto Mode

Since you haven't told us much to go on about how you have your your camera's AF settings set up, I'm going to go out on a limb here and guess that you are trying to use 'AI Servo AF' and are expecting to get a green focus confirmation light in the lower right hand corner of the viewfinder? When you don't get an AF confirmation light you assume the camera is not autofocusing properly?

When you are using the Green Square A+ selection on the exposure mode dial to put the camera in "Full Auto", the camera ignores your AF mode setting and uses whatever it thinks is best based on the information it's receiving from the light meter and PDAF focus array. Typically this will be 'AI Focus AF'. When you switch to Manual Exposure Mode, the camera will use 'AI Servo AF' if that is the AF mode you have selected.

The reason the focus confirmation light does not function in AI Servo AF mode is because the camera never stops tracking the subject and adjusting focus as necessary. The green focus confirmation light in the lower right corner of the viewfinder is an indicator that AF has been locked and has stopped measuring focus. That's the last thing you want to happen in AI Servo AF mode! With a moving target the green light wouldn't guarantee your subject is in focus, it would only confirm your subject was in focus at the distance it was from you when the AF was locked.

The AF confirmation light will only appear using 'One Shot' AF mode or 'AI Focus' AF mode before the camera automatically switches to 'AI Servo AF' mode if it senses the subject is moving. Canon EOS DLSRs offer three basic AF modes:

• AI Servo AF, which continuously tracks the subject until the shutter button is fully pressed.
• One Shot, which focuses the subject and then "locks" the focus position at that distance until the image is taken. In some scenarios, depending upon several other AF and Drive Mode settings, the camera may keep focus locked at a specific distance over multiple frames if the shutter button is never fully released past the "half-shutter" position or the 'AF ON' button is continuously held down during continuous burst shooting.
• AI Focus which is a hybrid of the other two and fairly unpredictable. It's supposed to start out in 'One Shot' mode and switch to 'AI Servo AF' if it senses the subject is moving.

With the appropriate menu selections with some (but not all) Canon models the viewfinder can be set up so that when doing tracking of moving subjects the focus points currently in focus are the only ones appearing on the LCD overlay visible in the viewfinder. Models that have such capability include the 1D X, 1D X Mark II and Mark III, 5Ds and 5Ds R, 5D Mark IV, and 7D Mark II. The 5D Mark III, which was released in 2012, does not feature an RGB+ir light meter and the processing power needed to run Canon's iTR (Intelligent Tracking and Recognition) AF in concert with the PDAF sensor. The 5Ds and 5Ds R added these features to the 5D-series in 2015, and the 5D Mark IV introduced in 2016 also has them.

Since your camera lacks the capability of the iTR AF, your best option is to do it the old fashioned way: use your eye pressed to the viewfinder to confirm your subject is in focus. Do note that if the camera can't focus on something it will warn you by rapidly flashing the green AF confirmation light. The more you practice it the better you will get at seeing what the AF system is really targeting and seeing when your subject is the center of focus.

For a much more comprehensive list of issues that may result in soft or out of focus images, please see this answer to How do I diagnose the source of focus problem in a camera?