Voadam
Legend
That is some stories about some specific fey.And of course the chaotic fey are well known to be unable to break their word and having to stick to their contracts!
I am unaware of that applying to satyrs or nymphs or dryads for example to name some common D&D fey. I am not sure core D&D fey have ever specified much on such binding rules or contracts. Usually it is 3rd party D&D sourcebooks that go into such and usually without specifics.
I do not think the vampire supernatural restriction on entering houses uninvited is an indication of lawful alignment or prohibits them from being chaotic under the various D&D alignment systems.
However if you wanted to portray fey as lawful, the supernatural rules and contracts and fey courts could be fey thematic roleplay hooks for a DM to emphasize from that perspective.