In a certain code language, 'PROCLAMATION' is coded as 161843121131203414, and 'MENACE' is coded as 13214132. How will 'LISTENER' be coded in that language?

Let's analyze the coding scheme by looking at the vowels and consonants separately:
Scribd

For MENACE → 13 2 14 1 3 2

M (Consonant) =13 (its normal alphabetical position)
Wikipedia

E (Vowel) =2 (Vowels are A=1,E=2,I=3,O=4,U=5)

N (Consonant) =14

A (Vowel) =1

C (Consonant) =3

E (Vowel) =2

Let's cross-verify with PROCLAMATION:

P=16

R=18

O=4 (vowel position)

C=3

L=12

A=1

M=13

A=1

T=20

I=3

O=4

N=14
Stringing them together: 161843121131203414. It matches perfectly.

Now apply this rule to LISTENER:

L=12 (consonant)

I=3 (vowel)

S=19 (consonant)
Filo

T=20 (consonant)
Filo

E=2 (vowel)

N=14 (consonant)

E=2 (vowel)

R=18 (consonant)

Combining all the numbers: 1231920214218.