Do I have to get a written agreement if...

The 1st Amendment gives you the right to link to other websites, and there's no copyright issue.

Also, linking isn't trademark infringement as long as it doesn't cause consumer confusion about the source or sponsorship of goods or services, and that means you have to be careful about using trademarks on your site. In addition, if you use the mnanufacturer's descriptions, you have to be concerned about making sure those descriptions are accurate and updated as necessary.

Consult a business and lawyer to help you form and structure the entity, get a business license and liability insurance, etc., secure an available and good business name, domain name, and trademark, etc.

Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.

Before you start an online business you need to learn how to do it. Visit amazon.com, query its books database for "online business," and buy a few books on the subject. After you read the books then ask your questions in a forum such as this.

Two points:

The thrust of your question seems to be whether it's lawful to publish links to any website -- but you preface your question by asserting that you intend to publish "affiliate links with third party vendors." That's confusing. The rules are, essentially, that (1) you may freely link to any web page if the content on that web page is lawful (no infringing content, no tools to facilitate unlawful conduct, no false or disparaging assertions, etc. etc. etc.) and (2) you may link to any web page owned by someone who "affiliates" with you but only according to the terms of the affiliate agreement you enter into with what you're calling the "third party vendor." So the short answer is that you may link to any lawful web page but you need a linking agreement if you create a relationship with a particular supplier.

As for copying others' product descriptions you may not lawfully do so if the description is sufficiently creative for copyright to attach. Some product descriptions are very creative, some are simply factual. You can copy the latter but not the former.

You need to hire an e-commerce attorney to help you set up your business. Good luck.