I'm a physician. Vitamin D deficiency is indeed associated with low mood and sleep issues, and has been correlated with other problems too, like worse COVID-19 outcomes. As you're suggesting, it's also common in wintertime when sunlight exposure drops due to shorter daylight hours and people hiding from the cold.

As a rule, though, I'd recommend against random supplementation where it can be avoided. If you have a doctor and have concerns, you'd be well served to discuss them before starting, as there are dozens to hundreds of other issues that can cause similar issues, including seasonal depression for those feeling sad and tired in winter. Vitamin D is one of the 4 major vitamins (along with A, E, and K) which is able to build up in toxic quantities in the body from excessive intake, and while this requires taking a lot of it, it's a possibility that shouldn't be ignored when you're considering your overall health plan