I've looked everywhere for guidance or some sort of procedures on this issue:

Governmental 401(a) plan is not a qualified retirement plan for Social Security purposes.
No Section 218 agreement in place.
FICA taxes therefore being paid on contributions to plan.

How might the plan be converted into a qualified 401(a) plan, which would then eliminate the FICA taxation of contributions? Can it be done? If so, what are the procedures for doing so? And what is the authority permitting this to be done?

Any assistance pointing me in the right direction would be much appreciated!

If it's a 401(a) plan as you aver, it is a qualified plan. If it is not a qualified plan, it is not a 401(a) plan. Care to re-ask your question? Is it:

How does a governmental entity convert its non-qualified plan to a qualified 401(a) plan? (It doesn't)

You refer to FICA taxation of contributions. Are those employer or employee contributions (or both)?

Generally, if those are employee contributions on which you wish to avoid taxation, the plan would need to be converted to a 457 plan rather than a 401(a) qualified plan.

Similarly, if they are employer contributions on which you wish to avoid taxation, a new profit sharing, money purchase or defined benefit plan which complied with 401(a) would have to be adopted. The old plan could be frozen or distributed, as desired.

No, thank you, vebaguru, but I am not asking about a non-qualified plan. I am asking about a qualified 401(a) plan that was treated as if it were not a qualified retirement plan for purposes of Social Security; i.e. FICA taxation. FICA taxes are being withheld from employer contributions. How might one take an existing 401(a) plan under such circumstances and treat it going forward as a plan that is a qualified retirement plan for Social Security purposes, so that FICA taxes are no longer withheld from the employer contributions?