The phrase at [one's] beckon call is an eggcorn derived from a mishearing of the at [one's] beck and call, which means freely available or ready to comply. The mistaken phrase is sort of understandable because someone who is at your beck and call is ready to be beckoned. Still, attentive readers are liable to see beckon call as wrong.
For example, a more careful editor could have saved these writers a little embarrassment:
The cast is rounded out by Selznick’s secretary Miss Poppenghul (Kelly Reeves), who’s at the producer’s beckon call for the entire five days. [Aurora Sentinel]
At beckon call, there is a bull or bear waiting to make his or her case on TV. [NASDAQ]
And the following writers use the unquestionable beck and call:
We turn on a tap, and behold! —safe, clean drinking water is at our immediate beck and call. [Tonic (link now dead)]
And with the Czech Philharmonic at his beck and call, Pallett has no trouble expanding the scope of his melodies far beyond any of his previous works. [Slant Magazine]