That’s because vieil is the masculine singular variant of the masculine singular vieux (un vieil ami, un ami vieux) but the masculine plural vieux (which is written exactly like its singular) has no alternate form so it can’t be but des vieux amis (resp. des amis vieux).

The feminine is much more regular: singular vieille, plural vieilles.

Short version: The root of this word ended in /z/ in plural and /l/ in singular. Both dropped off in non-liaison environments, but liaison preserved the final consonant of a word.

Long version: In general, adjectives that precede nouns participate in liaison. A consonant is inserted (or you could say a silent consonant is pronounced) before a vowel.

You could frame this question one way and ask how that consonant is selected. Then you could observe that there is no consonant when there’s no liaison; that it’s /z/ for plural; and that it’s something else ("elsewhere case") for singular:

While this shows some consistency, you might notice that there are two different cases here. For the singular with liaison, the first two examples above select a consonant that’s already present in the spelling, which makes sense. But the second two, vieux and beau, select an /l/, as if from the feminine! Where did they get that? I assume this is where your confusion comes from.

To answer this, you could frame the question the other way and ask how the final consonants disappeared in the first place. To start with the Latin forms:

• vieux is from vĕclus (from a diminutive form, whence the /l/)
• beau is from bellus (/l/ already in the root)

So the /l/ is there from the start; those unexpected vieil and bel forms are actually closer to the originals than vieux and beau.

We notice that all final consonants here dropped off when there was no vowel after them, and we notice that the liaison environment somehow yielded /z/ in plural forms but /l/ in singular forms. That can be explained by the use of the accusative form as the basis for noun forms in French. Accusatives in Latin mostly had /s/ endings in plural but not in singular (compare this chart).

Finally, you’ll notice that there is still an /l/ in the plural in Latin, just before the /s/, so where did it go in vieux? The answer is that as consonants softened and disappeared at the ends of words, only the last one tended to be saved by liaison (a rule still present in French today). Usually these are still around as silent consonants. However, with /l/, there was an intermediate stage where it sounded like /u/ and that’s where it was when the spelling happened to solidify. Many, many instances of the letter ⟨u⟩ in a final cluster are original /l/s.