You have several pitfalls awaiting you.
Firstly, as you've already discovered, the backslashes in path names cause problems in the strings, as they could also indicate C# escape sequences. It's usually good practice to use C#'s @"..." syntax for file names, partly to avoid needing to double up the backslashes and make it easier to read, and partly because you could inadvertently leave a \t in there and it'd go unnoticed for ages.
Secondly, the single-parameter call to Process.Start only takes a command - it cannot accept command arguments - so you have to call the two-parameter overload.
Thirdly, the quotes around of the value of the records argument need handling so that the C# syntax knows what you want with them - i.e. to pass them to the command. I've separated out the command arguments into two parts to make that clearer. I have elected to use backslashes to escape them, though using the alternative @"...""..." would be just as good, and the choice is largely down to personal preference unless the context points you strongly one way rather than the other.
string cmd = @"C:\Program Files\GoToTags\GoToTags Encoder\GoToTags.Encoder.exe";
string url = "http://petshop.intato.com/index.php?id=" + TxtBoxIDCode.Text;
string cmdArgs = "--records \"{'Url':'" + url + "','RecordType':'Website'}\"";
Process.Start(cmd, cmdArgs);
[edited to add:]
If for some reason you find you either want or need to use string.Format to help build your cmdArgs, there's a fourth gotcha waiting in the wings for you, in that string.Format looks for the brace ({ and }) characters to delimit insertion parameter specifications, but your records command-line argument wants braces characters in the string. The way to achieve that would be to double up the braces that you want, like this:
string cmdArgs =
string.Format("--records \"{{'Url':'{0}','RecordType':'Website'}}\", url)";
