This depends on your state's unemployment law and is not really a tax question. Sometime this income only counts if it exceeds a certain amount and some states have programs that continue your unemployment if you are trying to start a business. The WI reply doesn't count for every state. In addition the Covid law might make unemployment requirements more lenient. I assume this writing gig will result in a 'Form 1099 being issued.
I would try to contact your unemployment office and see what they say. Or say nothing and in the interim perhaps you can pursue this writing gig and be sure it will generate income for you and report you earnings when they materialize so you don't muddy the water and collect your unemployment benefits ASAP. If you have to repay them because your earnings from writing are too high bless you - you found a career. The W-9 doesn't go anywhere but to the firm that pays you. It means they are planning to send you a 1099 on the income you earn. That gets reported to the IRS and goes on your tax return. I hate cheaters so don't become one. Once you are collecting unemployment follow the requirements for reporting income. All of their requirements are usually available on their web site.
