Is it possible for a familiar to attune to and then wear or wield a magic item?
Yes, familiars are creatures and, as such, can attune to magic items so long as they meet the requirements for attunement (For example, the magic item cannot require the attuner to be of a certain class, and the DM must deem the familiar capable of wearing/attuning to the particular piece of equipment).
However, some additional restrictions may apply. For example, if you are playing in D&D Adventurer's League, you are subject to additional restrictions as per the latest DDAL FAQ:
Can my familiar attune to a magic item?
Any item attuned to a creature under your control (familiars, beast companions, etc.) counts against both your character’s limit of three attuned items and the character’s permanent magic item count.
This rule doesn’t imply that such creatures have the ability to attune to magic items. Whether or not a mindless undead creature— for example— can attune to something is subject to DM discretion.
As this is an official restriction for sanctioned play, it is not unheard of for such restrictions to also be used in home games.
Yes, familiars can do this.
Familiars are creatures, and I found no rules specifically forbidding familiars from using or wearing magic items.
It's worth noting that other restrictions might prevent familiars from using some items. For example, many familiars, especially the ones from the find familiar spell, cannot speak, and therefore cannot use command words.
Seeing a common theme of "familiar" subjects. You need to distinguish between Find Familiar spell, Warlock Pact Boon - PofC, and Variant:Familiar.
Find Familiar, offers you a spirit in the form and stats of a basic tiny creature. How does one of these creatures "attune" to something. I would argue that it isn't capable of any forms of higher thought regarding magic items and their purpose.
Attuning to an item requires a creature to spend a short rest focused on only that item while being in physical contact with it (this can't be the same short rest used to learn the item's properties). This focus can take the form of weapon practice (for a weapon), meditation (for a wondrous item), or some other appropriate activity. (DMG pg. 138)
PotC - Offers a higher intelligent spirit form, but like AL FAQ, attunement is linked to PC limits because of the spells link connecting you.
In addition, if the familiar should die from being sneezed at, it would drop the magic item and lose attunement as well.
Variant:Familiar being this is the actual creature and not a spirit manifested one, it should be more than capable to attunement. In addition, it wouldn't effect your attunement limits. Although, I would be weary of giving valuable objects to a creature who can willingly leave you and take your stuff.
I would say that familiars can benefit from passive magic items if they can meet the requirements for use or design. Nothing is stopping you from making "Horseshoes of Speed" as 8 little bracelets for your spider familiar.
Anything beyond RAW is up to your specific DM, as usual.