Hello Elizabeth, You can use 'would' here. It is used to express also something one intended to do. And in this sentence the person intended not to forget this past wonderful day.

Elizabeth, You can use both here. It depends on what you are trying to say. It was a wonderful day that I would never forget. ( Here the speaker is talking about the past. It is a look at the future starting from a time in the past.) example My first birthday party was a wonderful day that I would never forget for the rest of my life. It was a wonderful day that I would never forget. (You can also use would when talking about your desire or inclination.) example You might forget that it was a wonderful day, but I never would. It was a wonderful day that I will never forget. (Here you are talking about an event in the past and making a prediction that you will never forget it in the future.) example My wedding last year was a day that I will never forget.

According to the past tense, you can only use would here as the only choice that is grammatically correct. However you can say it is the day I would/will never forget. I mean you can say it is the day I loved but you can't say it was the day I love.