Yes, depending on how the spell targets creatures
Rubiksmoose gave an excellent break down of what an attack is here, but what's relevant to this question can be found on p.193-194, PHB:
Whether you're striking with a melee weapon, firing a weapon at range, or making an attack roll as part of a spell, an attack has a simple structure
If there’s ever any question whether something you’re doing counts as an attack, the rule is simple: if you’re making an attack roll, you’re making an attack.
As a familiar is not able to attack (excluding a Warlock Pact of the Chain familiar), but there is no mention to a familiar's ability to cast a spell, then the familiar should be able to cast any spell that does not require an attack roll.
The other question that may come up is the familiars ability to complete the components of the spell, but as answered here in regards to magic items casting spells, DMG p.141 states:
The spell is cast at the lowest possible spell level, doesn't expend any of the user's spell slots, and requires no components, unless the item's description says otherwise.
The bead created from Arcane Abeyance is a magical item, and would follow the rules of magic items from above, meaning the components of the spell would be irrelevant to the familiar's ability to cast the spell.
In regards to the concentration of the spell in question, Arcane Abeyance states, as you've linked:
...A creature holding the bead can use its action to release the spell within, whereupon the bead disappears. The spell uses your spell attack bonus and save DC, and the spell treats the creature who released it as the caster for all other purposes.
The only places the spell would not use the familiars abilities are on the spell attack bonus (irrelevant for a familiar) or spell save DC. All other aspects of the spell would be the responsibility of the familiar, including concentration.
