No. There is no problem with programming in Judaism.
As far as mimicking G-d's creation, we do that all the time: all craftsmen create things. In fact, we are commanded to mimic G-d in certain ways:
Leviticus 19:2 from Mechon Mamre.
-קְדֹשִׁים תִּהְיוּ: כִּי קָדוֹשׁ, אֲנִי יְהוָה אֱלֹהֵיכֶם.
-Ye shall be holy; for I the LORD your God am holy.
And thanks to DoubleAA for reminding me about:
Deuteronomy 28:9 from Chabad.org
יְקִימְךָ יְ־הֹוָ־ה לוֹ לְעַם קָדוֹשׁ כַּאֲשֶׁר נִשְׁבַּע לָךְ כִּי תִשְׁמֹר אֶת מִצְוֹת יְ־הֹוָ־ה אֱלֹהֶיךָ וְהָלַכְתָּ בִּדְרָכָיו:
The Lord will establish you as His holy people as He swore to you, if you observe the commandments of the Lord, your God, and walk in His ways.
Now, remember that human beings cannot really mimic G-d's creation at all. G-d creates from nothing (ex nihilo) while we simply modify and work with things that already exist.