There are multiple ways you can do this. If you want to know the quickest way to check on multiple YouTubers/Instagrammers at the same place, look for the last paragraph.
Check the description of the videos
because there is a lot of information available there. Some YouTubers also specify their business emails in their description box. Also if you find it there, they are ready to promote your brand yet there is a check here, he or she is desperately looking for collaborations.
About section of the channels
on Youtube where you can find information such as their business emails and websites. Some YouTubers will also provide their contact numbers which makes your job much simpler.
Commenting options
YouTubers or their team usually go through their comments for promotions/collaboration kinda queries. Though getting to know their timing and comment as early as possible to get higher chances of visibility. Comment precisely what you want to convey and give contact information such as mail and phone number.
Social media such as FB. Insta and Linkedin
Most of the YouTubers provide URLs for their social media profiles in the description. You can find much important information such as their mail, website, and phone number(not in most case). Famous YouTuber profiles can be found easily with a simple google search itself.
It may come in handy to contact them, but most YouTubers will get too many messages from their followers. To stand out in the crowd and make them read your proposal message might be difficult.
Email Requests
After getting their email manually from any of the methods above. Sending them an effective mail is a viable option. Tell them what you want to convey in short and precise yet convincing them to call you back. Don’t forget to provide your contact information and company details.
Other third-party tools
Online services like Influenex might be another option but it isn't free and also you can send invitations only. This is not an option I would suggest, as most of your efforts may go in vain because not every Youtuber there is for promotion.
The easiest way to connect with any YouTuber for paid collaboration
Now coming to the easiest yet most powerful way to directly contact Youtuber for promotion is through Pickmyad.com. There you find 100s of YouTubers who are ready to promote your product in their upcoming videos.
In pickmyad, you will find YouTubers from various niches, posting their Ad Slot (to promote your product) and opening for their next video. Select the YouTuber in your mind or choose from your niche. Added only in Pickmyad you can see their costing to promote your product, compare and choose. You can even select what type of ad you want the YouTuber to promote your product.
The best part, you can directly chat with them inside the platform. Know your collaboration progress, update reports and chat support incase a Youtuber ghost you!
So my suggestion will be to use pickmyad - select the YouTuber and book your slot using payment online. I hope, am clear in explaining. If you find this useful, share it with whom you think needs this information.
