If you can solve a problem they have, they will make time for you.
Most buyers care about three things: Cost, Productivity, Revenue. Think "CPR". If you can reduce costs, increase productivity, or drive additional revenue, they should want to meet.
Skip email or surveys. You want to talk to them and ask open ended questions about their needs to see if you can help.
On the call. Don't sell them anything. Ask them questions about what they are currently doing, and what solutions they would like to see for the "problem." Even if you just talk to 1 out of 20, listen to what they want and tweak your product to make them happy. If that customer is happy, then there is a good chance that other users will want what you've built. "Do things that don't scale" - PG.
Your goal on the call is to get that person using your product. Offer to do it for them. Write code for them. Help them implement your solution. Treat it like a consulting gig. Whatever they are looking for, you can do it.
A LinkedIn add compliments a direct email. But, a LinkedIn message is often ignored.
If you don't have the customer's email use Rapportive -- http://rapportive.com/
Rapportive is amazing! You add Rapportive to gmail. It is a free add-on for Firefox, Safari, Mailplane, and Chrome. You can guess at the email you are looking for...firstname@companyname.com, firstname.lastname@companyname.com, firstinitiallastname@companyname.com, etc. When you get a hit the right hand side of gmail lights up with the person's picture and links to their publicly available social networks, all of which gets displayed alongside the message. The sidebar replaces Google’s contextual ads, so that’s a double win.
Even if you don't like my advice you should checkout Rapportive, it helps.
PS: Try doing this at scale rather than relying on those 20. For example, twitter follow a few hundred people in the target market and then DM the ones that follow you back, asking for their advice.
Emails and phone calls are easy to ignore. I would first use LinkedIn to approach them asking for advice (keep the intro short, sweet, but intriguing). Then email a few times via LinkedIn - and THEN ask for a phone call.
If you do decide on phone calls, I would first find another group of 20 people you don't care as much about their answers- and use them first to "refine your approach" and reduce nerves.
Ask them for advice around the thing you are working on. Your goal at this stage should be to find 10 users who love your product.
A survey, in particular, seems awful for this.