Computing sums in some CSV files horizontally and in other CSV files vertically
Where we improve the existing code
First of, let's start by talking a bit about some issues in your code. We'll see if we can factorize some bits latter.
The first thing to note is that you iterate twice over your field_names to construct its final form and initialize self._totals. There is no need for it:
field_names = reader.next()
for i, code in enumerate(field_names):
    code = code.strip(string.punctuation).upper()
    field_names[i] = code
    if (len(code) &lt;= size) and (code not in bad_codes):
        self._totals[code] = 0

Next, we see that you differentiate the behaviours of your sums whether self._totals[code] has been initalized or not. One way to simplify it is to turn it into a collections.defaultdict. To be more precise, you will have to define self._totals = defaultdict(int) instead of self._totals = dict(). This will allow you to write, in compute_totals_v:
for row in reader:
    # get totals
    code = row[code_field].strip(string.punctuation).upper()
    self._totals[code] += string_utils.to_int(row[est_field])

This will also allow you to write, in compute_totals_h:
for row in reader:
    # get totals
    for code, val in zip(field_names, row):
        if (len(code) &lt;= size) and (code not in bad_codes):
            self._totals[code] += string_utils.to_int(val)

eliminating the need to initialize self._totals fields. Also, using zip here is more idiomatic. And since you are using Python 2 (I guess reader.next() would have been next(reader) in Python 3) you can use itertools.izip to avoid building a new list in memory.
We can see here that we're moving the test that initialized self._totals[code] into the “get totals” loop. But it is not efficient since we are re-doing it for each row of data. Time to improve that filtering by getting something closer to the if code in self._totals we had. The easiest way is to turn invalid codes into None and check if code is not None instead:
def compute_totals_h(self, size, bad_codes):
    with open(self._in_file_path, self._FILE_READ_MODE) as f:
        # initialize reader
        reader = csv.reader(f)
        field_names = [c if (len(c) &lt;= size) and (c not in bad_code)
                       else None
                       for c in (
                           code.strip(string.punctuation).upper()
                           for code in reader.next())]

        for row in reader:
            # get totals
            for code, val in zip(field_names, row):
                if code is not None:
                    self._totals[code] += string_utils.to_int(val)

    self._write_totals()

We now have something closer to compute_totals_v and can start to work on factorizing.
Where we get to do what you ask for
The general layout you extracted for your compute_totals method is quite close to what compute_totals_v and compute_totals_h has become. You should note, however, that compute_totals_h needs to return both reader and field_names for its initialization. So we need to return a dummy value (let's say None) in the initialization function for compute_totals_v as well. And thus, we need to accept a second parameter (the dummy value or field_names) in get_totals for both methods.
We also need to define the helper functions as inner functions of the method so we can capture the parameters of the method once and for all without having to pass them around.
# Somewhere, where you define self._totals:
# self._totals = defaultdict(int)

def compute_totals_h(self, size, bad_codes):
    def _init_reader(file_handler):
        reader = csv.reader(file_handler)
        field_names = [c if (len(c) &lt;= size) and (c not in bad_code)
                       else None
                       for c in (
                           code.strip(string.punctuation).upper()
                           for code in reader.next())]
        return reader, field_names

    def _get_totals(row, field_names):
        for code, val in zip(field_names, row):
            if code is not None:
                self._totals[code] += string_utils.to_int(val)

    self.compute_totals(_init_reader, _get_totals)


def compute_totals_v(self, code_field, est_field):
    def _get_totals(row, *args):
        code = row[code_field].strip(string.punctuation).upper()
        self._totals[code] += string_utils.to_int(row[est_field])

    self.compute_totals(lambda f: (csv.DictReader(f), None), _get_totals)

def compute_totals(self, initialize_reader, get_totals):
    with open(self._in_file_path, self._FILE_READ_MODE) as f:
        reader, field_names = initialize_reader(f)

        for row in reader:
            get_totals(row, field_names)

    self._write_totals()

I'm not convinced, however, of the improvement of this change. Readability seems worse to me. And calling the get_totals function for each row has its slight overhead.