If an airplane is already equipped for RVSM flight, how do we obtain a Letter of Authorization (LOA) for RVSM in order to actually fly in RVSM airspace?
You can find lots of information on RVSM at this FAA website.
From This MSWord document found on that site:
Operational Approval or Authorization. The FSDO or CMO will grant operational approval or authorize the operator to conduct RVSM operations by issuing Operations Specifications paragraphs B046 and D092 to air carriers or issuing a Letter of Authorization (LOA) to general aviation operators, as appropriate. Individual operator RVSM-compliant airframes are listed in D092 or on the LOA, as appropriate. The FSDO or CMO will take this action after it has determined that the operator’s aircraft are RVSM compliant; it has approved the operator’s RVSM maintenance program; a CMO has approved or a FSDO has accepted the operations programs and the office has accepted the operator’s plan for participation in the monitoring program.
This Excel spreadsheet lists points of contact at various FSDOs.