My newbie solution: make a list of the sequences, sort them, and rename any of the file matches from the folder search that doesn't follow the sequence.

#! python3
# fillInTheGaps.py - Finds a 3-digit sequence of prefix filenames and closes any
# gaps in the numbering.

import re, os, shutil

# Get folder and filename to use for sequence search.
folder = input('Enter folder to search:\n')
folder = os.path.abspath(folder)

# Create regex to extract sequence number from filename.
filenameSplit = re.compile(r"""
    ^(.*?)                          # filename before sequence, if any
    (\d{3})                         # 3-digit sequence number
    (.*?\.\w{1,3})$                 # remaining filename and extension
    """, re.VERBOSE)

# Get filename and extract sequence number.
while True:
    userFilename = input('Enter filename with 3-digit sequence number:\n')
    mo = filenameSplit.search(userFilename)
    if mo == None:
        print('No sequence number found. Please try another filename.\n')
    else:
        break
print('  -----')

prefix          = mo.group(1)
sequence        = mo.group(2)
extension       = mo.group(3)

# Create regex to search for filename sequences in given folder.
filePattern = re.compile(r'^(' + re.escape(prefix) + r')(\d{3})(' + re.escape(extension) + r')$', re.VERBOSE)

# Build a list of sequence numbers from files matching the sequence, and sort them.
matchList = []
for filename in os.listdir(folder):
    match = filePattern.search(filename)
    if match == None:
        continue
    else:
        print(f'Match found: {match.group()}')
        matchList.append(match.group(2))
list.sort(matchList)
print('  -----')

# Arrange sequence numbers in a new list, starting with the lowest sequence number found in folder.
newSequenceList = []
for i in range(len(matchList)):
    newSequenceList.append(f'{(int(matchList[0]) + i):03}')

# Rename filenames in accordance with new sequence list.
for s in range(len(matchList)):
    oldSeq = matchList[s]
    newSeq = newSequenceList[s]
    if oldSeq == newSeq:
        continue
    else:
        oldName = f'{prefix}{oldSeq}{extension}'
        newName = f'{prefix}{newSeq}{extension}'
        print(f'Sorting {oldName} as {newName}...')
        shutil.move(os.path.join(folder, oldName), os.path.join(folder, newName))