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.