Unpopular opinion, but yes, as his parent you can force him to have this, in the same way you could force him to have an operation or other medical treatment. His health is ultimately your responsibility.
I also didn't want the vaccine when I was at school, due to being scared of needles and injections. Didn't matter what my mum said in terms of the medical consequences of not getting it, because I knew and understood that, but was just very scared.
She said that if I refused it at school, she'd take me to the GP to have it done, and would hold me down if necessary and it'd be worse for everyone.
If this is the 3 injections in a year one, I believe she also said if she had to do that she would come into school for me to have the other two, and would do what was necessary for me to be given it, whether that was holding me down, or giving me it herself (she's a nurse).
I got it in school (all 3), had a friend hold my hand, took some sweets for afterwards, and was generally alright. Plenty of others who were nervous too.
I'd go for the tact of giving him a choice with both options being acceptable to you, along the lines of 'Would you rather have it at school with your friends, or at the GPs with me on a different day'. If he's still saying no, 'No isn't an option, you have to have the vaccine, your choice is about where you'd like to have it, at school or the doctors'.
If he's refusing to make a choice, sign the school forms anyway, let them know he's a bit nervous and that you'd like him to have it done, but to please let you know if they don't manage to give it to him or he refuses, and then you'll take him to the GPs for it.