We have not been able to find an explicit source on this common practice (although a practice often counts as a source in and of its own right). However, there seems to be a similar case, discussed by the poskim, which can serve as a precedent.
The Taz[1] forbids using one sefer to raise up another one to make it easier to read, as he is using something holy for a use for which “wood or stone” would work just as well. (It is permitted, according to the Taz, to place one sefer on another if the bottom sefer was initially brought to the place to be used for learning, not as a stand.) It would seem, then, that one should not use the siddur to kiss the sefer Torah, as he can use his hand or tallit, etc.
On the other hand, the Magen Avraham[2] disagrees with the Taz and allows one to bring one sefer in order to prop up another. His main source is the gemara[3] that states that one may move a bima to a place where its presence will prevent tuma (impurity) from entering a beit k’nesset. One can understand the Magen Avraham in a limited manner: the holy object may be used only if the reason it is moved is not obvious or only if it is in a stationary position when used. However, it appears from the context and language of the Chayei Adam[4] and Mishna Berura[5] that they understand the Magen Avraham in an inclusive manner that would apply to our case as well. In other words, if the use is not, in and of itself, degrading and, in fact, is intended to aid in a mitzva-oriented activity, it would be permitted.
It is also possible that the Taz would agree that his comparison to “wood or stone” is not appropriate in our case. The cogency of the argument may depend on the circumstances. Some may use the siddur as a convenient way to extend the hand and not bother to get closer to the sefer Torah. If so, its use is for human convenience, even though it is within the context of a positive religious experience (showing love for the Torah). However, many use the siddur because they feel it is more respectful for the Torah to be touched by something holy, and not by human hands. From that perspective, the siddur is not a replacement of wood or stone but an aid to show respect for the sefer Torah. Moreover, since the sefer Torah is on a higher level than the siddur, the grounds for leniency are stronger than in the Taz’s case.
This concept has precedent in a similar case. The Sha’arei Ephraim[6] describes that one who gets an aliya kisses the Torah with his tallit or the me’il (the cover of the sefer Torah). He does not stipulate that this practice is permissible only if there is no alternative. Thus, considering that the me’il has a higher level of kedusha than a talit or even a siddur, it seems that one can freely use holy things for kissing a sefer Torah. Perhaps one could argue that only those items, like a me’il, whose kedusha stems from its use to serve the sefer Torah, may be used to serve the sefer Torah in any way.
In summary, it seems there is a strong basis to permit using a siddur to kiss a sefer Torah, even if alternatives exist. Therefore, one should not criticize or discourage those who have the practice to do so. On the other hand, we cannot totally refute the contention that it is improper to use the siddur for that purpose.