No, the spores will not stack.
In the DMG on page 2521, "Combining Game Effects" it says:
Different game features can affect a target at the same time. But when two or more game features have the same name, only the effects of one of them—the most potent one—apply while the durations of the effects overlap. For example, if a target is ignited by a fire elemental’s Fire Form trait, the ongoing fire damage doesn’t increase if the burning target is subjected to that trait again. Game features include spells, class features, feats, racial traits, monster abilities, and magic items. (DMG 252)
Thus, a PC will only experience the effects one spore at a time no matter how many times they are hit with the ability.
This is also backed up in the PHB section on conditions which says:
If multiple effects impose the same condition on a creature, each instance of the condition has its own duration, but the condition's effects don't get worse. A creature either has a condition or doesn't. (PHB 240)
Since Spore inflicts the poisoned condition, it cannot be applied again to make the poisoning worse. In this case, there really is no duration so applying it again just does nothing.
So, no matter how many times you get hit, if you fail your save you take 1d10 damage at the start of each of your turns.
1 - this was not present in early printing of the DMG and was added as errata which you can find here. Subsequent printings do include the new text.