No, you cannot cast spells requiring a material component while wild shaped
As noted in the question, Beast Spells states in part:
You can perform the somatic and verbal components of a druid spell while in a beast shape, but you aren't able to provide material components.
I read the intent of this as allowing spells with S and or V components, and disallowing those with M components. This is based on my reading of the various ways a caster can satisfy the Material Components requirement of a spell, as elaborated below:
The rule describing druidic foci on PHB page 151 says:
A druid can use such an object as a spellcasting focus, as described
in chapter 10
Chapter 10 rules on material components (PHB p. 203) give 3 options for satisfying the requirement to provide material components for a spell that has a material component requirement -- find the specific listed items, use a component pouch (assumed to have all the right stuff in it), or use a spellcasting focus:
A character can use a component pouch or a spellcasting focus
(found in chapter 5, "Equipment") in place of the components specified
for a spell.
(exceptions for costly material components follow)
Normally a caster would not actually find the specific listed components (except in the case of costly components) -- but this does leave an option for a caster separated from his/her spellcasting focus or components pouch if they can find the right stuff in their environment. The "typical" ways of satisfying the material components for spells that require non-costly components are to use either a pouch or a focus.
Since Beast Spells states that you cannot provide material components, you cannot provide that required spell component any of the listed ways. This is most likely because you do not have hands, as further down that section it states:
A spellcaster must have a hand free to access the spell's material
components -- or to hold a spellcasting focus -- but it can be the same hand that he or she uses to perform somatic components.
[Note, the explicit reference to the spellcasting focus here occurs only in the second printing of the PHB]
Thus you require a hand free to either access a component pouch or to hold the druidic focus. Other than primates, a beast form will not have hands at all. A special rule lets you do the gestures without (humanoid) hands, but specifically says you cannot provide the material components.
... well, maybe, IF your beast form has hands...
Although I've hypothesized that it's about usually not having hands, there's nothing in the rule that says you can't provide material components unless you have hands, so for the few beasts that have hands, while I think a DM could rule this either way, I would probably still rule "no" -- all three methods are listed under "material components" in the PHB and provide a variety of ways of satisfying the "material components" requirement of the spell, thus any of them count as "providing material components". The rule that says you cannot provide material components does not actually say that it is because you do not have hands, that is just my guess as to why that rule might make sense.
However, if you did interpret that rule as meaning that only providing actual listed components is prohibited, and that using the alternate methods of satisfying the material components requirement were permitted, then if your beast form did indeed have a hand with opposable thumbs, I could see allowing it.
This all really comes down to whether "provide material components" is meant to include all the ways one can satisfy the material component requirement of a spell or only some. A DM who wants to rule on this has to decide:
Is using a focus a different type of action than providing material components, or is it a different way of providing material components?
Until WOTC weighs in on this, I don't think the PHB specifies clearly which of these interpretations applies, so this comes down to DM discretion.
If you rule (as I would) that "provide material components" means doing any of the things listed in the Material Components section of the PHB, then you can't do it in beast form, period. You can't do it by finding the components in the forest, by using a material components pouch, or by using a spellcasting focus, no matter how many hands you have.
If you rule that "provide material components" means providing the listed components, but that the prohibition on providing material components doesn't apply to satisfying the material components requirement of the spell with a druidic focus, then your only obstacle is that a spellcasting focus requires a free hand to use; thus while in a beast form with hands, you could cast with a druidic focus.