I see that your vector file is a .ai file. .ai is a file format that is proprietary to Adobe Illustrator, and if we don't have Illustrator, then we can't open your file.
You'll want to use the .svg file format instead for vector graphics. It is open source and is easier to share on the internet. To do this, open your file in Illustrator and then go to File --> Save As...
In the "File Type" box, click on the arrow and select .SVG from the drop down list, then save it as a svg. Once it is saved as a svg file, you should be able to upload it onto a image hosting website.
Now in your web browser go to an image hosting website (I suggest Tinypic.) Click on "Choose File" and select the svg file you saved earlier, then click on "Upload Now!". It will take a minute or so to upload it (after you fill in the captcha), and then spit out a link that you can copy and paste here for us to see.
As for the file size of your image, it is possible that the canvas size of your vector file is too large. There should be an option to compress a vector file in Adobe Illustrator, but I don't know where it is.