Two questions get mixed up here: whether you need an emergency action plan at all, and whether it has to be on paper.
Whether. An employer must have an emergency action plan whenever an OSHA standard in Part 1910 requires one (29 CFR 1910.38(a)). The most common trigger is the portable fire extinguisher standard: an employer that provides extinguishers but does not expect employees to use them, and evacuates instead, is relying on an EAP. Others include process safety management, HAZWOPER, and several Subpart Z standards.
Whether on paper. An emergency action plan must be in writing, kept in the workplace, and available to employees for review. However, an employer with 10 or fewer employees may communicate the plan orally to employees (29 CFR 1910.38(b)). The ten counts employees, not full-time equivalents, and the oral option does not remove any of the plan's required content — it only changes the medium.
Contents. At a minimum: procedures for reporting a fire or other emergency; evacuation procedures including type of evacuation and exit route assignments; procedures for employees who remain to operate critical operations before evacuating; procedures to account for all employees after evacuation; procedures for employees performing rescue or medical duties; and the name or job title of every employee who may be contacted for more information (29 CFR 1910.38(c)). An employee alarm system with a distinctive signal is required ((d)), and employees must be designated and trained to assist in an orderly evacuation ((e)).
Review with employees. The employer must review the plan with each covered employee when the plan is developed or the employee is initially assigned to a job, when the employee's responsibilities under the plan change, and when the plan is changed (29 CFR 1910.38(f)). There is no annual review requirement in the text; the triggers are assignment and change.
A plan that lives in a binder nobody can find fails the "available to employees" test as surely as no plan at all.