Husk, 18 17 14 13 bytes

Edit: -3 bytes, and then another -1 byte, thanks to Leo

€¹mΠ†dfṁ→m½Pd

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Outputs nonzero integer if it's a vampire number, zero otherwise.

€¹                  # index of input if present, zero otherwise, in
  mΠ                # products of each element-pair
    †d              # combining digits as a number from
         m½         # first & second halves of
           P        # all permutations of
            d       # digits of input;
      f             # and filtering only element-pairs for which
       ṁ→           # sum of last digits is nonzero

Husk, 18 17 14 13 12 bytes

Edit: -3 bytes, and then another -1 byte, thanks to Leo, and -1 byte thanks to inspiration from pajonk's R answer (3rd edit)

€¹mΠ†dm½f→Pd

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Outputs nonzero integer if it's a vampire number, zero otherwise.

Commented penultimate version:

€¹                  # index of input if present, zero otherwise, in
  mΠ                # products of each element-pair
    †d              # combining digits as a number from
         m½         # first & second halves of
           P        # all permutations of
            d       # digits of input;
      f             # and filtering only element-pairs for which
       ṁ→           # sum of last digits is nonzero

Husk, 18 17 bytes

€¹moΠmdfo→Fz+m½Pd

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Outputs nonzero integer if it's a vampire number, zero otherwise.

€¹                  # index of input if present, zero otherwise, in
  moΠ               # products of each element-pair
     md             # combining digits as a number from
             m½     # first & second halves of
               P    # all permutations of
                d   # digits of input;
       f            # and filtering only element-pairs for which
        o→Fz+       # last element of zipped sums is nonzero

Husk, 18 17 14 13 bytes

Edit: -3 bytes, and then another -1 byte, thanks to Leo

€¹mΠ†dfṁ→m½Pd

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Outputs nonzero integer if it's a vampire number, zero otherwise.

€¹                  # index of input if present, zero otherwise, in
  mΠ                # products of each element-pair
    †d              # combining digits as a number from
         m½         # first & second halves of
           P        # all permutations of
            d       # digits of input;
      f             # and filtering only element-pairs for which
       ṁ→           # sum of last digits is nonzero

Husk, 18 17 bytes

€¹moΠmdfo→Fz+m½Pd

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Outputs nonzero integer if it's a vampire number, zero otherwise.

€¹                  # index of input if present, zero otherwise, in
  mΠ                # products of each element-pair of
          mom       # results of mapping these functions:
             d      # combine digits in
              ½     # first & second halves
                    # across all elements of:
               P    # all permutations of
                d   # digits of input;
   f                # and filtering only element-pairs for which
    F¤|%10          # one or other is nonzero modulo 10

Husk, 18 17 bytes

€¹moΠmdfo→Fz+m½Pd

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Outputs nonzero integer if it's a vampire number, zero otherwise.

€¹                  # index of input if present, zero otherwise, in
  moΠ               # products of each element-pair
     md             # combining digits as a number from
             m½     # first & second halves of
               P    # all permutations of
                d   # digits of input;
       f            # and filtering only element-pairs for which
        o→Fz+       # last element of zipped sums is nonzero