The text is unclear, leaving room for exploits.
As you've said, there's no clear explanation on what happens to the equipment when the steed vanishes. While the heavily implied solution would be that it also vanishes alongside the steed, the rule text as written doesn't mention it at all.
Let's analyse it a bit further. The one condition given for the equipment to vanish is the following, emphasis mine :
Any of the equipment created by the spell vanishes in a puff of smoke if it is carried more than 10 feet away from the steed.
As a side note, if we twist the text even further, the equipment could technically be more than 10 feet away from the steed while the steed is still active, as long as it isn't carried. For example, placing it down on the ground, then taking the steed somewhere else wouldn't make it vanish. Picking it up at any point while the steed is away would instantly make it vanish though.
Back to our main question. As written, the equipment won't vanish as long as it isn't being carried away from the steed. So then, what about when the steed vanishes? The condition for the equipment vanishing is a distance condition. However, the steed simply stops existing. The summon isn't an entity or creature from another plane, so they don't even return to some other plane when the spell ends, they simply disappear.
As there is no definition of "distance with something that doesn't exist", the text can be interpreted in many ways :
- Distance is infinite : carrying the equipment will make it vanish.
- We use the last position of the steed : you can move the equipment in a 10-feet circle around this position.
- The distance is undefined : you can carry the equipment anywhere without it vanishing.
Of course, this isn't intended.
It's up to the DM to decide of what interpretation they want to go with. The most natural, and very likely intended, effect is that the equipment vanishes alongside the steed, but if a DM wants to rule otherwise using the above, they're free to do so.