Yes - if you use Wild Shape first, and dependent on the beast form chosen
As per the Player’s Handbook, when you use Wild Shape:
Your game statistics are replaced by the statistics of the beast, but you retain your alignment, personality, and Intelligence, Wisdom, and Charisma scores. You also retain all of your skill and saving throw proficiencies, in addition to gaining those of the creature.
You don’t retain any other statistics, so if you had used the Shifting feature beforehand you would lose the bonuses and new attacks. But you can still use features granted by your race and class as long as “the new form is physically capable of doing so”.
While the Shifting description says you “assume a more bestial appearance”, that is only descriptive - the specific effects for each Shifter type could easily be applied to a normal animal if they have appropriate anatomy to modify, making them take on bestial features they don’t normally have. Arguably a housecat that suddenly sprouts large fangs would have a more bestial appearance than a regular cat!
Looking at what those features specifically are, most of them are easily explicable for many kinds of beasts:
- Beasthide Shifters get temporary hit points and a bonus to AC - that’s easily explained for most Wild Shape forms by growing a tougher hide, feathers, scales etc. Soft-bodied creatures like an octopus or jellyfish would be harder to justify, but perhaps not - an octopus could grow an external shell like a nautilus, perhaps?
- Longtooth Shifters get an additional unarmed strike with elongated fangs, which they can use as a bonus action. This could easily be applied to any animal with teeth, or if your DM is generous, any kind of mouth (imagine a raven sprouting fangs). This could upgrade the attack of a small animal or one without a specific bite attack.
- Swiftstride Shifters get an increase to their walking speed and the ability to move a short distance as a reaction to enemy movement. This could apply to any animal with a walking speed, by growing longer legs or becoming leaner. Conversely “a more bestial appearance” wouldn’t reasonably include growing legs on a fish or make sense for some body plans with very slow walking speeds (like an octopus or snail).
- Wildhunt Shifters get advantage on Wisdom checks and prevent enemies from gaining advantage against them. That’s easily represented as improving the senses of any animal, as most beasts only get advantage on Wisdom (Perception) checks for specific senses at best.
So depending on the Shifter type, the Shifting benefits should be “physically possible” for many if not most Wild Shape forms, adjudicated on a case-by-case basis.