The idiomatic phrase is ce n’est pas l’envie qui m’en manque (where m’ is indirect object in the dative case), and it means not that I don’t want to or I’d love to, but… (Larousse). I think en is very legitimate here, in that it explains "what kind of desire you are not lacking", and that is normally expressed in the surrounding context of the phrase. Consider:

[Y retourner], ce n’est pas l’envie qui m’en manque. J’aimerais tant aller me recueillir sur la tombe de mes parents, restés là-bas. (Centre Presse)

It is obvious to me that en replaces here d’y retourner. So we could rephrase:

Ce n’est pas l’envie d’y retourner qui me manque.

However, it is true that the recent tendency is to remove "en" altogether, even with the risk of "envie" being misunderstood as "envy". The use of the phrase is idiomatic, so people will recognise it even without "en". You can even find such an example in some dictionaries:

Ce n’est pourtant pas l’envie qui manque (Vogüé,Morts, 1899, p. 338). (LaLangueFrancaise)

So there you go, this is the original phrase (with en), but feel free to omit the "en" if it confuses you. Just know that this would be a grammatically inaccurate use of the phrase, which is however acceptable due to its wide use.

By looking at the following transformation one can get a clearer picture; "V" means "verbe", "XXXX" is a subject, "YYYY" is the predicate less the verb.

XXXX V pas YYYY → Ce n’est pas XXX qui V YYYY

Three examples

[L’été] n’est pas [la saison la plus clémente de l’année].
Ce n’est pas [l’été] qui est [la saison la plus clémente de l’année].

[Les enfants] ne jouent pas [avec les grandes personnes].
Ce ne sont pas [les enfants] qui jouent [avec les grandes personnes].

Ceux-ci ne se placent pas sur une case de cette couleur.
Ce ne sont pas [ceux-ci] qui se placent [sur une case de cette couleur].

The pattern of transformation should be clear. Applied to the sentence under scrutiny it gives what follows, where the pronoun "leur" has been replaced by a noun because the particular placement of this pronoun makes the transformation more difficult.

• Ce n’est pas [le désir de qqc] qui [en manque à ces personnes].
  [Le désir de qqc] ne manque pas [en à ces personnes].

It is clear that "en" is a dangling element in the sentence. In other words this is a set locution that is not analysable, and as such it is worthless because we can say simply "Ce n’est pas le désir (de qqc) qui leur manque" or if considered as non-idiomatic there is an error that was never detected.

This can be seen as follows too.

The relative pronoun "qui" is subject of "manquer"; as this pronoun has "désir de qqc" for antecedent you have to conclude that what is not lacking is "le désir de qqc".

• Ce n’est pas le désir (de qqc) | le désir de qqc leur en manque.

If you choose "de qqc" as the antecedent, which is not exact anyway, you have "manque de qqc". This is wrong, it means nothing; therefore it could be "désir de qqc", and this gives what follows.

• Ce n’est pas le désir (de qqc) | le désir de qqc leur [du désir de qqc] manque.

• Ce n’est pas le désir (de qqc) | le désir de qqc à eux manque [du désir de qqc].

This is nonsense.

This pronoun "en" is too much here; it should have been removed long ago.

If the complement of "désir" is not considered ellipted then "en" can be used, but in the first clause, the main clause.

C’est le temps des vacances et ils ont la possibilité de partir en voyage ; ce n’en est pas le désir qui leur manque.