Your car should have stop/start and will have a capacitor. The battery for these cars is different to a normal battery as is the charging system.
It is most likely to be the battery but before buying a new one I would check the electrolyte levels then disconnect the battery (both terminals) and give it a long charge preferably with a charger that has some ‘intelligence’. You may consider taking it out and giving it a gentle shake during charging to get the electrolyte moving around between the plates. Once the charger has indicated that it is full put it back in the car and see if it has ‘recovered’.
The ‘correct’ battery for these cars appears to be only available from Mazda at a ‘reasonable’ cost. If you do a search there is another thread regarding this. Nd1 12v battery needed? - Technical Area / Electrics & ICE - MX-5 Owners Club Forum (mx5oc.co.uk)
If you do need a new battery and decide not to use the Mazda one please do come back on here and tell us if it worked out OK.
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