"Some magic items allow the user to cast a spell from the item. 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."
No, not if YOU are casting the spell
The restrictions for Wild Shape state the following (PHB, p. 67, bold added):
You can’t cast spells, and your ability to speak or take any action that requires hands is limited to the capabilities of your beast form.
Note that the ability to cast spells is ruled out unconditionally. If they has said "your ability to cast spells or speak or take any actions that requires hands ..." you would be able to cast spells in Wild Shape if your shape could perform the required components, or if there were none.
As it stands, even if the spell requires no components, you still cannot cast it.
With certain magic items, you could still gain the effects of spells
Note that there are some magical items which create the effect of spells without you being the one that cast them. Consider the potion of Heroism (DMG, p. 188, bold added):
For 1 hour after drinking it, you gain 10 temporary hit points that last for 1 hour. For the same duration, you are under the effect of the bless spell (no concentration required). This blue potion bubbles and steams as if boiling.
So while Wild Shaped into a beast with sufficient manual dexterity, you could drink a Potion of Heroism, and be under the effect of the Bless spell since it wasn't you who cast it.
Contrast this with the Helm of Telepathy (DMG, p. 174, bold added):
While wearing this helm , you can use an action to cast the detect thoughts spell (save DC 13) from it.
In this case, you are explicitly the one casting the spell ("from it" does not change this fact). So you could not do so in Wild Shape.
Naturally, once you hit level 18 in Druid, you would be quite capable of doing so. But not before.
You can't cast a spell from an item in beast form, nor use anything that requires speech or a body part that your beast form doesn't have.
Wild shape says:
You can't cast spells, and your ability to speak or take any action that requires hands is limited to the capabilities of your beast form.
According to the DMG, items activate by one of four methods:
Wearing and wielding items
A magic item meant to be worn must be donned in the intended fashion. ... In most cases, a magic item that's meant to be worn can fit a creature regardless of size or build.
You can wear any item that your beast shape has the right body parts for. You need feet to wear boots, hands to wear gloves, and so on.
A command word is a word or a phrase that must be spoken for the item to work.
No. You can't speak in beast form, so you can't use a command word item.
Some items are used up when the are activated.
You can use any consumable that you have the right body parts to activate. If you can physically drink a potion, you can use it. You can't speak to activate scrolls.
Some magic items allow the user to cast a spell from the item.
No. You can't cast spells, because wild shape specifically says you can't cast spells. Helm of telepathy is one such item.