There's an approved nasal vaccine in the United States (FluMist). It's use was not encouraged for a while because it didn't vaccinate against H1N1 (recent years have included H1N1). It doesn't look to be particularly more effective than the shot (marketing focuses on no shot, not on it working better; the CDC says the effectiveness is similar).

There's multiple candidate intranasal SARS-CoV-2 vaccines, just none that have concluded trials.