Searching online English dictionaries
Looks pretty good. I have a few suggestions though:


There should be a way to exit (maybe if you don't type anything when it asks which dictionaries?)
You should probably use urllib.parse.urljoin instead of +.
You should probably use recursion instead of using while True then always breaking or continueing.
You don't need parentheses when returning tuples.
Bare raise statements are for reraising exceptions. Otherwise, it gives you RuntimeError: No active exception to reraise.
Don't use bare except.
I would probably use a list of tuples instead of using zip.
You could pass around the whole list instead of its length.
You can use enumerate(iterable) instead of zip(range(len(iterable)), iterable). What enumerate does is create an iterable that when iterated, yields tuples of increasing numbers starting with 0 and the next item from the iterable (in your case a list) you are enumerating. More info at the Python Docs.
You should handle the logic for * in one place.
Contained the logic for handling numerals within the get_user_data function.
You should probably name your main function main instead of __main__.
Don't combine multiline strings and '\n'.
It might be good to have it display the result directly in the console instead of the web browser, but that would be a whole other project.
You should probably return directly instead of using break.
I might have made a few other minor changes.


Note that some of the changes are just personal preferences.

The result (also available at https://repl.it/@solly_ucko/DictionarySearchCodeReview):

import webbrowser
from urllib.parse import urljoin

class OnlineDictionary:
    """Represents an online dictionary.

    Attributes:
name: str
url: str
    """
    def __init__(self, name, url, sep):
        """Initializes an OnlineDictionary object.

        name: str
        url: str
        sep: str
        """
        self.name = name
        self.url = url
        self.sep = sep

    def search(self, query):
        """Searches the dictionary for a given query.

        url: str
        query: str
        """
        query = self.sep.join(query.split())
        webbrowser.open(urljoin(self.url, query))


def print_instructions(message, instructions, dictionaries):
    """Prints the available dictionaries, their corresponding
    numerals and the instructions.

    message: str
    instructions: str
    names: list
    num_of_dics: int
    """    
    print(message)

    for numeral, (name, _, _) in enumerate(dictionaries):
        print(numeral, name)

    print()
    print(instructions)     


def get_user_data(dictionaries):
    """Asks the user for which dictionary 
    to use and for what to search for.

    num_of_dics: int

    Returns: tuple
    """

    while True:
        to_use = input('Specify which dictionary or dictionaries to use.\n')

        if to_use == '':
            return (), ''
        elif to_use == '*':
            to_use = dictionaries
        else:
            try:
                to_use = to_use.replace(' ', '').split(',')
                numerals = []

                for num in to_use:
                    num = int(num)

                    if num not in range(len(dictionaries)):
                        raise ValueError

                    numerals.append(num)

                to_use = set(map(dictionaries.__getitem__, numerals)) # set is used to get rid of duplicates
                query = input('Type something to search for.\n')
                return to_use, query
            except ValueError:
                continue


def main():
    dictionaries = [
        ('Merriam-Webster Dictionary', 'https://www.merriam-webster.com/dictionary/', '%20'),
        ('Oxford Dictionary', 'https://en.oxforddictionaries.com/definition/', '_'),
        ('Camberidge Dictionary', 'https://dictionary.cambridge.org/dictionary/english/', '-'),
        ('Collins Dictionary', 'https://www.collinsdictionary.com/dictionary/english/', '-'),
        ('Longman Dictionary of Contemporary English', 'https://www.ldoceonline.com/dictionary/', '-'),
        ('The American Heritage Dictionary of the English Language', 'https://ahdictionary.com/word/search.html?q=', '%20')
    ]

    message = 'These are the available dictionaries:\n'

    instructions = """To use a dictionary type its corresponding numeral.
To use more than one dictionary separate the numerals with ",".
To use all of them type "*". Do not type the quotation marks.
"""

    print_instructions(message, instructions, dictionaries)

    while True:        
        to_use, query = get_user_data(dictionaries)

        if not to_use or not query:
            break

        print()
        chosen_dictionaries = [OnlineDictionary(name, url, sep) for name, url, sep in to_use]

        for dictionary in chosen_dictionaries:
            print('Searching %s for "%s"...' % (dictionary.name, query))
            dictionary.search(query)

        print('Finished searching. Check your Web browser.\n')

if __name__ == '__main__':
    main()