At the moment, it's not entirely clear.
The Marvel Fandom page for the MCU version of Dr. Strange states that he is currently the Sorcerer Supreme:
Doctor Stephen Vincent Strange M.D., Ph.D is the Sorcerer Supreme and a Master of the Mystic Arts. Originally a brilliant, although arrogant, neurosurgeon, Strange got into a car accident which resulted with his hands becoming crippled. When all Western medicine failed him, Strange embarked on a journey that led him into Kamar-Taj where Strange had made the discovery of magic and alternate dimensions, being trained by the Ancient One. Though focused on healing his hands and returning into his career, Strange learned more of the mystic arts and helped the Masters prevent Kaecilius from merging the Earth with the Dark Dimension, but not before witnessing the Ancient One's death. Following the demise of their mentor, Strange became the protector of the New York Sanctum in New York as well as the Earth's protector from any new inter-dimensional threats.
That in itself doesn't prove anything, as Marvel Fandom is not an official source. However, the page does offer a citation from an official source, that being Dr. Strange's bio on Marvel.com, which states the following:
Formerly a renowned surgeon, Doctor Stephen Strange now serves as the Sorcerer Supreme—Earth’s foremost protector against magical and mystical threats.
This statement appears above the entries for both the MCU and comics versions of Strange.
Given that this page combines entries for both versions though, I'm not convinced that the Sorcerer Supreme statement applies specifically to the MCU version.
If you look further down the entry for the comics version, it again states that he's "Earth's Sorcerer Supreme", whereas the entry for the MCU version just says that he's "Earth's newest Master of the Mystic Arts".
And the In Comics Full Report section documents the specific circumstances in which the comics version of Strange became Sorcerer Supreme, whereas the On Screen Full Report section says nothing about the MCU version receiving that title.
Also, as noted in TheLethalCarrot's answer, Wong stated at the end of Doctor Strange (2016) that Earth had no Sorcerer Supreme at that time.
Wong: You might have a gift for the mystic arts, but you still have much to learn. Word of the Ancient One’s death will spread through the Multiverse. Earth has no Sorcerer Supreme to defend it. We must be ready.
Doctor Strange (2016)
It's possible that MCU Strange became Sorcerer Supreme some time after this, but it seems odd that Marvel would choose to have such an important moment for the character happen entirely off-screen, and neglect to even mention it in passing in any of the movies.
It's also worth bearing in mind that the comics version of Dr. Strange didn't become the Sorcerer Supreme until a full decade after his first appearance. He debuted in Strange Tales Vol. 1 #110 in July 1963, but didn't inherit the title of Sorcerer Supreme from the Ancient One until Marvel Premiere #10 in September 1973.
Marvel Premiere #10
The latter issue also featured the first confrontation between Dr. Strange and Shuma-Gorath. It's rumoured that Shuma-Gorath will appear in the upcoming Doctor Strange in the Multiverse of Madness (2022), and if so, that would seem like the perfect time to have Dr. Strange become the Sorcerer Supreme within the MCU.
We'll obviously have to wait and see if that's what will actually happen though.