Rare problem... access the pump (under the package shelf) and check to see if it's getting 12 volts while cranking. If so, replace the pump. If not, start checking relays, etc. Could be the computer is not closing the relay because it doesn't know the engine is cranking (bad cam position sensor, etc.) I'd guess it's not the pump (15 years Miata specialist).
The 1999 Mazda Miata is a returnless fuel system. It has a Fuel Pressure Regulator in the tank (above the fuel pump), and a snub at the end of the fuel rail.
You can get a 1999 Mazda Miata fuel cap at any auto parts store.
The 2002 Mazda MX-5-Miata runs on regular unleaded.
The 2004 Mazda MX-5-Miata runs on regular unleaded.
The 2005 Mazda MX-5-Miata runs on regular unleaded.
The 2003 Mazda MX-5-Miata runs on regular unleaded.
The 2001 Mazda MX-5-Miata runs on regular unleaded.
in the tank
The 2007 Mazda MX-5-Miata runs on premium unleaded (required).
The 2010 Mazda MX-5-Miata runs on premium unleaded (required).
The 2013 Mazda MX-5-Miata runs on premium unleaded (recommended).
The 2011 Mazda MX-5-Miata runs on premium unleaded (recommended).