I'm assuming that the invitations have already been sent? Because it would have been perfectly appropriate to have put the dress code on there. "Cocktail attire" or similar.
But yes, if you are getting a lot of questions, I'd definitely put it on the website. If you want to qualify it you can say, "We've been getting a lot of inquiries as to dress code for the wedding. It's cocktail attire. We're looking forward to celebrating with our friends and family."
Or you can just put it as a simple line.
"Wedding will be cocktail attire."
If I didn't see it on the invitation and saw it on the website, I wouldn't think you didn't trust me to dress appropriately, I'd assume you'd just forgotten to put it on the invitation because regardless of the venue or time of day, that's always the first question I have -- what to wear. Now, yes, sometimes, it's obvious. Evening wedding with full dinner at a hotel, it's gonna be dress up time. Wedding on the beach at 10 am, definitely more casual, i.e., I wouldn't assume shorts but definitely sandals and no hose. But the best way to remove all doubt is just to state your intentions clearly. It's not only acceptable, it's expected.