In order to match up the different addresses to the one company name, you need to give each company a unique identifier. You might not want to use a number with 15,000 entries but maybe a combination of the letters of the company names. With so many companies a number may be the only real solution, if you are willing to choose a six digit number. You can copy the structure of the table that houses the company names and add an auto number field (the auto numbers can be fudged to start at any number, but I don't know off the top of my head).
Once the auto number field is set the number you want, append all the names of the company to this new table. Now you must add the unique identifier of the name of the companies to the address table. Link the two tables by their numbers and you won’t have a problem with multiple addresses.
I hope you already have a form where the company names are listed and a subform where the company addresses can be seen, added or edited.
I hope this will give you an idea of what can help you. You can also look at the Northwind files included in Access since they usually have very good examples. And before I forget, just in case, I always created a new database just for testing, not just a new table or form.