Although it could be considered ironic that Helen desperately wants the doll to have eyes because Helen doesn't have eyes that function in the first place, I think the more important irony here is dramatic irony.  Dramatic irony, of course, is when the audience knows something to be true that other characters do not know.  Notice the following stage directions:

Helen then yanks at her aunt's dress, and taps again vigorously for eyes.

Aunt Ev:  What, child?

Obviously not hearing, Helen commences to go around, from person to person, tapping for eyes, but no one attends or understands.

This is a classic case of dramatic irony.  Here we share in the truth with Helen.  She knows these "things" on people's faces are of some important use, ... something she does not have.  She wants this so desperately that she insists her new doll have the same "ability."  Only the audience is in league with Helen here.  Although WE "attend and understand," the entire family certainly does not.  Therefore, the following reaction from Helen surprises them:

Helen is back at Aunt Ev, fingering her dress, and yanks two buttons from it. ... Helen pushes the buttons into the doll's face.  Kate now sees, comes swiftly to kneel, lifts Helen's hand to her own eyes in question.

And so, as Kate adds eyes to the dolls face (in final understanding), the issue is resolved.

Finally, you use the word "panic" in your question.  This baffles me.  Helen doesn't panic here, ... she is reacting out of frustration.  Not only is she frustrated that her doll doesn't have eyes but also she is frustrated that no one in her family understands her desire.  In fact, it is the second fact that sends her begging for attention.  Is ripping the buttons off of an aunt's dress actual "panic"?  Hmmm, probably not.  If there is any panic at all, it is a panic of misunderstanding.

Poor Helen.  Well, lucky for her, things are about to change!  Enter:  Annie Sullivan.