The box is grounded via conduit
The locknut on the right of your second photo along with the lack of ground wires and the doubled white wire running into the right-hand locknut all combine to say one thing: this is a conduit job, and since the box is metal and no ground wire is present, the conduit must be metal as well to provide the grounding path.
As a result, NEC 404.9(B) point 1 applies, rendering the switch yokes grounded:
(B) Grounding. Snap switches, including dimmer and similar
control switches, shall be connected to an equipment grounding conductor and shall provide a means to connect metal faceplates to the equipment grounding conductor, whether or not
a metal faceplate is installed. Metal faceplates shall be
grounded. Snap switches shall be considered to be part of an
effective ground-fault current path if either of the following
conditions is met:
(1) The switch is mounted with metal screws to a metal box
or metal cover that is connected to an equipment
grounding conductor or to a nonmetallic box with integral means for connecting to an equipment grounding
(2) An equipment grounding conductor or equipment bonding jumper is connected to an equipment grounding
termination of the snap switch.
If your new switch provides a ground wire...
If your new switch has a ground wire on it, that can be terminated to a 10-32 machine screw threaded into the mating hole on the back of the box. (It's visible top right in the second photo, just to the left of a larger hole.)