I did a little research for you. Here it is: This is the Missouri Law but you can probably apply in either state and it will be recognized in the other. When I took my goddaughter I just did it in my state because it was easier.
In your state it is the Probate Court that handles it. There are plenty of forms on the internet for cost but I bet the Clerk of the Probate Court has free forms.
475.060. Application for guardianship — petition for guardianship may be
limited to school registration or insurance coverage. —
Any person may file a petition for the appointment of himself or some
other qualified person as guardian of a minor or guardian of an
incapacitated person. Such petition shall state:
(1) The name, age, domicile, actual place of residence and post office
address of the minor or incapacitated person if known and if any of these
facts is unknown, the efforts made to ascertain that fact;
(2) The estimated value of his real and personal property;
(3) If the minor or incapacitated person has no domicile or place of
residence in this state, the county in which the property or major
part thereof of the minor or incapacitated person is located;
(4) The name and address of the parents of the minor or incapacitated
person and whether they are living or dead;
(5) The name and address of the spouse, and the names, ages and addresses
of all living children of the minor or incapacitated person;
(6) The name and address of the person having custody of the person of
the minor or incapacitated person;
(7) The name and address of any guardian of the person or conservator of
the estate of the minor or incapacitated person appointed in this or any
other state;
(8) If appointment is sought for a natural person, other than the public
administrator, the names and addresses of wards and disabled persons for
whom such person is already guardian or conservator;
(9) In the case of an incapacitated person, the fact that the person for
whom guardianship is sought is unable by reason of some specified physical
or mental condition to receive and evaluate information or to communicate
decisions to such an extent that the person lacks capacity to meet
essential requirements for food, clothing, shelter, safety or other care
such that serious physical injury, illness or disease is likely to occur;
(10) The reasons why the appointment of a guardian is sought;
(11) A petition for the appointment of a guardian of a minor may be filed
for the sole and specific purpose of school registration or medical
insurance coverage. Such a petition shall clearly set out this limited
request and shall not be combined with a petition for conservatorship.
(RSMo 1939 §§ 377, 447, A.L. 1955 p. 385 § 293, A.L. 1983 S.B. 44 & 45,
A.L. 2000 S.B. 944)
Prior revisions: 1929 §§ 377, 448; 1919 §§ 373, 444; 1909 §§ 405, 474
(1957) Where application for guardianship of minors did not give the
domicile of the minors or of their parents, nor the names and addresses of
the parents or spouses of the minors nor state who had custody of the
minors, it failed to confer jurisdiction on the court to appoint the public
administrator as guardian. In re Dugan (A.), 309 S.W.2d 145.