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UPDATE: I tried to implement the defaultdict solution, but got an error when trying to update the field with the conctenated values. Here's the updated code, and the error I got is below that:

# Insert Search cursor on a feature class or table to iterate through row objects and extract field values.
# Sort values of a Search Cursor based on the CaseField and ReadFromField in ascending order.
# Define what will happen once the curser moves through each row.
# While it is in each row it will get the value of CaseField field that you are using as id to iterate.
# While it is in row it will also get the value of the ReadFromField field that you want to concatenate.
# Set the value of the dictionary to the values read by the cursor from the ReadFromField.
# Set an if condition for what should the cursor do when it reads through fields with same ID or the CaseField value.
# In if condition set the new value to last value of the ReadFromField + the defined delimiter + the new value that is read.
# Again set the dictionary value to this new value.
# Set the loop to have the lastid to the id that you got from getValue before it goes through the seconnd loop and so on...
# Set the loops last value variable to the last value that was read such that it starts with that last value for the second loop and so on...


cur1 = arcpy.SearchCursor(InputTable, "", "", "", CaseField +" A;" + ReadFromField +" A")

for row in cur1:
    id = row.getValue(CaseField)
    value = row.getValue(ReadFromField)
    dictionary[id] = value        
    if id == lastid:
        value = str(lastvalue) + Delimiter + str(value)
        dictionary[id] = value
    lastid = id
    lastvalue = value


# Delete cursor and row objects to remove the lock on the data that will remain until either the
# script completes or the cursor object is deleted. 
del cur1, row

    # Insert Update cursor to update or delete rows on the specified feature class, shapefile, or table. 
    # Define what will happen once the curser moves through each row.
    # While you are in each row set the cursor to get the value of the CaseField that is used as Id to iterate.
    # Set the value of the field that the concatenated values should be written to with the dictionary values that you concatenated in the code above.
    # Set the cursor to update the row values with the dictionary values.

    cur2 = arcpy.UpdateCursor(InputTable) 
    for row in cur2:
        id = row.getValue(CaseField)
        row.setValue(CopyToField, dictionary[id])
        cur2.updateRow(row)

Error Info: <type 'exceptions.ValueError'>: Row: Invalid input value for SetValue