Convert *.VCF contact file for cellphone import
Your function generate_vcard_contact_string obviously belongs to the Contact class. It even takes a Contact instance as the first argument.

I would actually elevate this to be the magic __str__ method, which is for example called when you do print(contact):

class Contact(object):
    """
    Contains the contact information,
    including a list of phone numbers.

    Arguments:
        object {[type]} -- [description]
    """

    def __init__(self):
        self.FirstName = ""
        self.LastName = ""
        self.Phonenumbers = []

    def __str__(self):
        """
        Generates the vCard string for this contact.
        Will generate a separate vCard for each phone number of the contact.

        Returns:
            string -- the generated vCard string
        """

        base = "\n".join(["BEGIN:VCARD",
                          f"N:{self.LastName};{self.FirstName}",
                          "TEL:{}",
                          "END:VCARD"])

        return "".join(base.format(number) for number in self.Phonenumbers)


You should also place your calling code under a if __name__ == "__main__": guard to allow importing from other scripts without executing the code. It is also complex enough to be encapsulated in its own function, IMO:

def parse_contacts(file_name):
    with open(file_name) as f:
        contact = None
        for line in f:
            # Some lines are build like "item1.TEL;...",
            # remove "item1.", "item2.", to ease parsing
            if line.startswith("item"):
                line = line[line.index(".")+1:]

            if "BEGIN:VCARD" in line:
                # Marks the start of a vCard,
                # create a blank contact to work with
                contact = Contact()

            elif line.startswith("N:"):
                # Line contains a name in the format N:LastName;FirstName;...;...;
                # Only LastName and FirstName will be used
                chunks = line[2:].split(';')
                contact.LastName = chunks[0].strip()
                contact.FirstName = chunks[1].strip()

            elif line.startswith("TEL:"):
                # Line contains a phone number
                # phone number type may be specified by "TYPE=...",
                # currently, TYPE will ne omitted
                # One contact may contain multiple phone numbers.
                number = extract_number(line)
                contact.Phonenumbers.append(number)

            elif "END:VCARD" in line:
                # Marks the end of a vCard,
                yield contact

def main():
    for contact in parse_contacts('contacts.vcf'):
        print(contact)

if __name__ == "__main__":
    main()


Your extract_number function can also be simplified using a single regex:

def extract_number(line):
    """
    Extracts the phone number from a vCard-file line,
    by removing everything but numbers and '+'

    Arguments:
        line {string} -- the line to extract the phone number from

    Returns:
        string -- the phone number
    """
    return re.match(r'TEL:(\+?\d*)', line).groups()[0]