Factorials and never ending cycles!

Jelly, 6 bytes

D!SµÐĿ
    ÐĿ  Repeat until the results are no longer unique. Collects all intermediate results.
D           Convert from integer to decimal (list of digits)
 !          Factorial (each digit)
  S         Sum

Try it online!

I don't see any other way to make it shorter other than to do as told.

Specs

• Input: 132 (as command-line argument)
• Output: [132, 9, 362880, 81369, 403927, 367953, 368772, 51128, 40444, 97, 367920, 368649, 404670, 5810, 40442, 75, 5160, 842, 40346, 775, 10200, 6, 720, 5043, 151, 122, 5, 120, 4, 24, 26, 722, 5044, 169, 363601, 1454]

Python 2, 88 bytes

import math
f=lambda x,l=[]:l*(x in l)or f(sum(math.factorial(int(i))for i in`x`),l+[x])

Try it online!

05AB1E, 12 bytes

[DˆS!O©¯så#®

Try it online!

Explanation

[               # infinite loop
 Dˆ             # add a copy of current value to the global list (initialized as input)
   S            # split current number to digits
    !O          # calculate factorial of each and sum
      ©         # save a copy in register
       ¯så#     # if the current number is in the global list, exit loop
           ®    # retrieve the value from the register for the next iteration
                # implicitly output the global list