Did George Muller have a Biblical basis for not accepting illegitimate orphans?

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According to The Life of Trust: Being a Narrative of the Lord's Dealings With George Müller, his autobiography:

I began the service of caring for children who are bereaved of both parents, by death, born in wedlock, and are in destitute circumstances, on Dec. 9, 1835. —Chapter XVII, "Reaping Bountifully", p. 294.

That, it seems, confirms the criteria he used. Later he explains his primary purpose in establishing a house for orphans:

This, then, was the primary reason for establishing the orphan house. I certainly did from my heart desire to be used by God to benefit the bodies of poor children, bereaved of both parents, and seek in other respects, with the help of God, to do them good for this life. I also particularly longed to be used by God in getting the dear orphans trained up in the fear of God; but still, the first and primary object of the work was, and still is, that God might be magnified by the fact that the orphans under my care are provided with all they need, only by prayer and faith, without any one being asked by me or my fellow-laborers, whereby it may be seen that God is faithful still, and hears prayer still. That I was not mistaken, has been abundantly proved since November, 1835, both by the conversion of many sinners who have read the accounts which have been published in connection with this work, and also by the abundance of fruit that has followed in the hearts of the saints, for which, from my inmost soul, I desire to be grateful to God, and the honor and glory of which not only is due to him alone, but which I, by his help, am enabled to ascribe to him.—Chapter VII. "Home for Destitute Orphans", p. 115.

In other words, his goals were in order:

Show that God answers prayers,
Do good for orphans in this life, and
Bring orphans to faith.

He further explains when considering the possibility of opening another house:

The many distressing cases of children, bereaved of both parents, who have no helper. I have received two hundred and seven orphans within the last sixteen months, and have now seventy-eight waiting for admission, without having vacancies for any. I had about sixty children waiting for admission about sixteen months since, so about two hundred and thirty children have been applied for within these sixteen months. But, humanly speaking, for the next sixteen months the number of applications will be far greater, as the work is now so much more widely known; except it be that persons may hear that the new Orphan House is quite full, and on that account may consider it useless to apply.—Chapter XX, "A New Victory of Faith", p. 366.

He goes further to explain the difficulty orphans who don't have influential sponsors have getting help from other charitable institutions for orphans. Because of this, Müller seems to have been willing to make certain exceptions:

I feel myself particularly called upon to be the friend of the orphan, by making an easy way for admission, provided it is really a destitute case.—Chapter XX, "A New Victory of Faith", p. 365.

Whether admitting a child born out of wedlock was one possible exception, I don't know.

Summary

Without further information, we can only speculate what his reasons were. Certainly the general attitude toward marriage and legitimacy played a part. But limited space and an abundance of applicants likely had a role as well.