Why not. There’s nothing to stop you applying to employers ‘on spec’ to ask if they have any opportunities for somebody with your skills.
Employers admire initiative, but don’t expect miracles. If you are an exceptional candidate, it’s possible an employer could create a role for you, even if they don’t have any current vacancies. But, it may be more constructive to see success in terms of getting them to consider you for future roles.
You can increase your chances of grabbing their attention by being clear and specific about how you can help them with their challenges and targets as an employer.
Emails are easily ignored and, besides, it shows courtesy and confidence to pick up the phone and give your prospective employer a call. Prepare your pitch in advance (don’t leave it to chance), and check before you launch into your spiel that they’re not in the middle of something. Then follow it up with a polite email with your CV attached to thank them for their time. Try to speak to the person most likely to be recruiting rather than delivering your speech to the office intern.
If you're dying to work for a certain company and can't find a way in, consider applying for a lower level job and working your way up. Once on the inside, you’ll be the first to hear about other jobs that take full advantage of your skills and talents.