Yes. GNOME has its bundled screenshot tool, and Zoom supports screen recording on GNOME/Wayland. Sway has grim + slurp for screenshots, and wf-recorder or wlrobs for screen recording.
However, one difference between X11 and Wayland, is that the role of the compositor (window manager) is much larger in Wayland. Thus, I believe it is up to the compositor to offer an API for features like this in Wayland, and the API may differ between compositors. In practice, GNOME has done their own thing; KDE and Sway collaborate on protocols, and are partly compatible; nearly every other compositor is based on Sway's wlroots, and should be compatible with Sway.
However, one difference between X11 and Wayland, is that the role of the compositor (window manager) is much larger in Wayland. Thus, I believe it is up to the compositor to offer an API for features like this in Wayland, and the API may differ between compositors. In practice, GNOME has done their own thing; KDE and Sway collaborate on protocols, and are partly compatible; nearly every other compositor is based on Sway's wlroots, and should be compatible with Sway.
[1]: https://github.com/swaywm/sway/wiki