Yes this will work but you have to check the monster's statblock
These features work together; this is because Reckless Attack does not require the Attack action but only an attack and attacks made will benefit from Reckless Attack because they are "melee weapon attack rolls" that are in the case of most beasts "using Strength".
Reckless Attack states:
When you make your first attack on your turn, you can decide to attack recklessly. Doing so gives you advantage on melee weapon attack rolls using Strength during this turn...
Notably this does not require you to take the Attack action, but only make an attack (See this Q/A if the distinction is confusing ("What does upper-case-A-Attack action vs. lower-case-a-attack mean?")). It also only benefits "melee weapon attack rolls using Strength".
The Multiattack feature on the giant badger monster states:
Multiattack. The badger makes two attacks: one with its bite and one with its claws.
Note, there is no order specified for the attacks (it does not say "one with its bite and then one with its claws) so you could use them in any order.
This is supported by Jeremy Crawford (lead game designer) in his tweet here so it is certainly the rules as intended:
Q. When a creature multiattacks, does it choose the order or go in the order listed (Seems alphabetical)
A. The DM decides the order of a creature's attacks using Multiattack, unless the creature's stat block mandates a particular order.
Clearly this was under the assumption that the DM is controlling the creature, and so the PC using Multiattack would likewise be able to choose the order of its attacks.
The badger's bite and claw attacks are both classified as "Melee Weapon Attacks". And the Player's Handbook's section on "Modifiers to the Roll" it states:
The ability modifier used for a melee weapon attack is Strength...
Unfortunately, this does not mean any monster with a melee weapon attack definitely uses Strength.
An example and explanation is below:
For example, the eagle's Talons attack has a +4 to hit even though its strength has a -2 modifier. The Eagle seems to be using Dexterity for its melee weapon attack. However, there is no explicit mention of this in any statblock, so the decision is ultimately up to the DM, though they can make an educated guess as to what the intention of the designer was.
You would have to determine what ability score modifier a creature is using for its attack, this can be done by first determining its proficiency bonus (look at a skill it is proficient in and subtract the ability score modifier normally associated with that skill to get the monster's proficiency bonus).
Then look at its bonus for the to hit part of that attack to see how much is added (or subtracted) from the proficiency bonus.
If a monster has equal strength and dexterity modifiers I would ask your GM to decide what is used.
Of note: some monsters are not proficient in all of their attacks and so they may not be adding their proficiency bonus to the to hit modifier.
An example of this is laid out in the Sage Advice Compendium:
Q. The bite attacks of ghouls and ghasts don’t appear to have proficiency bonus added in. Intentional or mistake?
A. Intentional; it’s a ghoul/ghast thing. They’re exceptionally bad at biting, compared to their claw attacks. Monsters sometimes have stat quirks like that.
When looking at the statblock runs into issues just ask your GM, you can even ask them before you get into the statblock and just use what makes sense to the GM.