If there is a thrift store close to you see if they have knitting needles... When I first started knitting I got alot of my needles at thrift stores...

Or go on line to
knitnet.com,
Lionbrand.com or
Michaels.com

and you can get needles there... yes you can do a pattern with smaller needles, but this is where gauge comes in... the smaller needles with give you a tighter fabric... try knitting a swatch with the yarn you want to use, and see if it comes out the way the pattern says... x stitches per row, x stitches per inch... Alot also depends on what you are making... dish cloths or the like are nice with a tighter fabric...

Hope this helps

yes it will look remotely the same, but the finished object will be smaller.

a size 7 needle is 4.5 mm, you might also try to make your own needle. find an appropriate sized wooden dowel. use a pencil sharpener to make one of the ends pointy. use some relatively fine grit sand paper to blunt the sharp end a bit and to make the needle nice and smooth. then rub a bit of candle wax all over (or rub some wax paper on it). voila! handmade needle for your handmade objects!

<<Is it possible to do the same patterns with a different size needle and come out with something that looks remotely like the same result (obviously with slightly different sized stitches)? >>

You could--with a lot of pattern-altering know-how and some algebra. But, the question is, why would you want to? It's easier to just spend a few bucks and buy the size 7. Sorry, but I could never understand those knitters that would come into a store and not want to buy a different size needle. If you knit, then you need needles, and they are different sizes for a reason. And you're going to need circulars of different length coils and double-points (and maybe straights if you, for some odd reason, prefer knitting with straight needles to circulars). Why make knitting so much harder and limit yourself to projects that only call for the one size needle you own? In fact, do yourself a favor and stock up on a wide range of needles sizes once you have the time and money or find a good sale on needles. Size US 5-9 in 24" coil should get you through most adult sweater projects, with cooresponding 16" or double-points for the sleeves.