We have similar interests then
You are miles ahead of me in terms of where I was in first year! Getting an internship in first year is just a bonus, generally its unnecessary and quite hard to obtain. Your best bet, as you're doing already, is just to contact local companies and offer unpaid work. If not, do some thorough research or learn some skills to make you shine in your interviews next year!
In your second year summer, look for something investment focused (preferably AM of course!) would be my advice. As you mentioned, I'm sure the big 4 hire a large amount from your uni as they do mine. It will serve as a stepping stone to hopefully obtaining an AM internship or something quite relevant in your 3rd year summer.
Skills you want to pick up for AM are generally quant based, depending on the role you enter. Maybe try reading some famous investment books over summer (The Intelligent Investor springs to mind), or learning about specific strategies such as value investing.
Just to note, as you can tell I am no expert. I'm simply stating what I would have done myself if I was at motivated as you in my fresher year (which I wish I was!).
Thanks, and well my CV is nothing impressive, but I guess I must've done at least something right... onwards and upwards, I'm very focused on achieving more and more now!