You can automatically save the time of randomization, or any other status change, using the @CALCDATE action tag. This tag lets you calculate a new date that is offset from an existing date. In this case, the existing date is now. The following text can be entered as the Action Tags/Field Annotation for a text box field type. The variable randomized_arm should be replaced with the randomization field as configured on the project, and randomized_date should be replaced with the variable name that is tracking the datetime of randomization.

@READONLY
@CALCDATE(if([randomized_arm] != '' and [randomized_date] = '', 'now', [randomized_date]), 0, 'd')