Man, I remember the trouble I had when I first tried contacts years ago. That reflex of closing your eye to protect it is very strong, but you can train to ignore/circumvent it.
I am concluding from your later remarks that you have disposable soft contacts. Good news is those should be easier to get used to than hard contact lenses.
Yes, I know how frustrating it can be not to get them in right away, but it is a question of practice and patience. So take a deep breath and don't panic.
I only have experience with hard contact lenses which I was told are healthier in the long run and last for years but are harder to get used to at first. Don't know how much of this can be applied to soft ones (which are easier on the eye at first but also bigger, so you have to open your eyes even more).
- calm down
- if you do this in front of the bathroom mirror over the sink, make sure the plug is down, so that if you should drop the lense it can't end up in the drain
- make sue the lense is moist enough, your index finger not only clean but dry, so the lense doesn't slid around on it (very annoying!)
- the lense has to be parallel to the eyeball otherwise you might hit the eyeball with the edge (maybe more a hard lense than a soft lense issue) which might trigger anotherb blink reflex
- you obviously don't want to poke your eyes out but I have noticed that I often underestimated the distance between the finger and the eye and moved the finger too softly towards the eye, this takes practice
The blinking reflex is natural and hard to get rid of. But you can do it and it gets easier every time.
Once you have the contacts in your eyes the next step is keeping them in! At least with the hard ones that too takes practice. You start with maybe 15 or 30 minutes the first day, a little longer the second day, etc. Because it stings a bit at first and feels uncomfortable to have them in. Also, you might initially run around with your eyes half closed and have a hard time opening them up like you would without lenses.
All of this is just for your information to have you mentally prepared. I tried contacts when I was 16 and went through a very tough time getting used to them. But they kept fogging up in my eyes. So I had to return to glasses. Then at 21 I tried it again and it worked just fine without any major trouble whatsoever.
I have now been wearing them for ten years. Putting them in and getting them out (with the help of a little rubber sucker for hard contacts, I never learned to do it with the fingers) takes only a few seconds and I don't even notice I am wearing them.
Contact lenses are a gift of civilization and I am very happy to have them. But getting used to them and learning how to use them can be a trial. Don't give up or alternatively try again later like I did. You are not alone with that problem and I think it is worth persisting.