Vectorize My Existing Logo

I have a logo only available as a JPEG/PNG and I need it converted into a true, clean vector. The final artwork must hold up everywhere I use it—YouTube banners and watermarks today, social-media avatars and website headers tomorrow, and even large-format printed pieces down the road... (Budget: $10 - $30 USD, Jobs: Adobe Creative Cloud, Adobe Illustrator, Adobe Photoshop, Graphic Design, Illustrator, Logo Design, Photoshop, Vector Design)