Find a word at the beginning of the line?

Not in the user interface itself. The only GREP codes for "beginning of ..." are \A for Beginning of Story and "^" for Beginning of Paragraph and "\<" for Beginning of Word – see also the Online Help at https://helpx.adobe.com/indesign/using/find-change.html#metacharacters_for_searching (actually that list is far from complete as "Beginning of Story" is missing; but there is no hidden "Beginning of Line" code).

It can be scripted, but for a script tailored for your needs we need more information. This quick short one, for example, will search only the first occurrence of a word from the start of a selected story, and I am pretty sure you want more than that.

findText = prompt ("Find", "");
if (findText)
{
	app.findGrepPreferences.findWhat = '\\<'+findText+'\\>';
	list = app.selection[0].parentStory.findGrep ();
	while (list.length)
	{
		item = list.shift();
		if (item.characters[0] == item.lines[0].characters[0])
		{
			item.select();
			exit();
		}
	}
	alert ("Not found");
}

 as it will only find the first occurrence of the input word/words at the beginning of a line:

Hi,

open find window (command/control F), select GREP tab and type:

^yourWord

^ – means shift 6 

regards M.