I was looking at Volo's Guide to Monsters, the section on "Monstrous Adventurers", and wondered whether you include a background when you use that to create a character of one of the species listed there. It doesn't say in the section, and there aren't any sample characters. I think that many of the backgrounds listed in the PHB wouldn't be applicable for some of the species.
There's nothing in Volo's that removes the background part of character creation.
Further, Volo's instructs the DM to "Work with the player to determine how the character ended up as an adventurer," and to consider the character's bond, trait, flaw, and ideal—all concepts that are associated with choosing a background.
It is true that "many of the backgrounds listed in the PHB wouldn't be applicable for some of the species." The following information from the PHB should be taken into account.
Your DM might offer additional backgrounds beyond the ones included in chapter 4, and might be willing to work with you to craft a background that's a more precise fit for your character concept.
—Ch 1: Step-by-Step Characters, p. 13-14
The sample backgrounds in this chapter provide both concrete benefits (features, proficiencies, and languages) and roleplaying suggestions. [emphasis added]
Customizing a Background
You might want to tweak some of the features of a background so it better fits your character or the campaign setting. […] If you can't find a feature that matches your desired background, work with your DM to create one.
—Ch 4: Personality and Background, p. 126-127
If you are a DM considering adding monstrous races to your character options, you should consider making up appropriate backgrounds or working with your players to do so following the guidelines in Ch 4. If you are a player in a campaign where monstrous races are allowed, you should discuss the possibility of designing a custom background with your DM.