Would a man be permitted to ask Hashem to bring about permitted sexual acts with his current or future wife as part of the requests he makes during the Amidah, given that we are generally highly discouraged from allowing sexual thoughts to enter tefillah? Would it make a difference if he was wearing tefillin while doing so, given that the prohibition of "impure thoughts" while wearing tefillin is generally held to be still more stringent?
The simple answer to the question in your title is yes. As Rambam explains in Mishnah Torah, Hilchot Tefillah v'Birchat Kohanim, Chapter 1, halacha 2:
וּלְפִיכָךְ נָשִׁים וַעֲבָדִים חַיָּבִין בִּתְפִלָּה לְפִי שֶׁהִיא מִצְוַת עֲשֵׂה שֶׁלֹּא הַזְּמַן גְּרָמָא אֶלָּא חִיּוּב מִצְוָה זוֹ כָּךְ הוּא שֶׁיְּהֵא אָדָם מִתְחַנֵּן וּמִתְפַּלֵּל בְּכָל יוֹם וּמַגִּיד שִׁבְחוֹ שֶׁל הַקָּדוֹשׁ בָּרוּךְ הוּא וְאַחַר כָּךְ שׁוֹאֵל צְרָכָיו שֶׁהוּא צָרִיךְ לָהֶם בְּבַקָּשָׁה וּבִתְחִנָּה וְאַחַר כָּךְ נוֹתֵן שֶׁבַח וְהוֹדָיָה לַה׳ עַל הַטּוֹבָה שֶׁהִשְׁפִּיעַ לוֹ כָּל אֶחָד לְפִי כֹּחוֹ:
The obligation in this precept is that every person should daily, according to his ability, offer up supplication and prayer; first uttering praises of God, then, with humble supplication and petition ask for all that he needs, and finally offer praise and thanksgiving to the Eternal for the benefits already bestowed upon him in rich measure.
All that one needs normally includes healthy sexual relations.
All that said, it doesn't eliminate the general idea of prayer with proper intention (or as you seem to describe, with pure thoughts). So no, you shouldn't indulge in sexual fantasies during that request. That isn't prayer, it's indulging your Yetzer HaRah.
And since it came up in the comments to your question, regular, proper, sexual relations with ones spouse are not only for procreation, nor limited to pleasure as you pointed out. But as Rambam states in Hilchot De'ot 3:2, it is also part of a normal, healthy regimen.
כְּדֵי שֶׁיַּבְרִיא וְיַעֲמֹד שָׁלֵם הוֹאִיל וְאִי אֶפְשָׁר לָאָדָם לִחְיוֹת אֶלָּא בַּאֲכִילָה וּשְׁתִיָּה. וְכֵן כְּשֶׁיִּבְעל לֹא יִבְעל אֶלָּא כְּדֵי לְהַבְרוֹת גּוּפוֹ וּכְדֵי לְקַיֵּם אֶת הַזֶּרַע. לְפִיכָךְ אֵינוֹ בּוֹעֵל כָּל זְמַן שֶׁיִּתְאַוֶּה אֶלָּא כָּל עֵת שֶׁיֵּדַע שֶׁהוּא צָרִיךְ לְהוֹצִיא שִׁכְבַת זֶרַע כְּמוֹ דֶּרֶךְ הָרְפוּאָה אוֹ לְקַיֵּם אֶת הַזֶּרַע:
that he may become healthy and stay healthy, seeing that it is impossible for man to sustain life save by food and drink. Likewise in connubial relation there shall be no coition save when it is of benefit to health, or to sustain species. One should, therefore, not yield to every urge, save when knowing that emission of semen is a helpful means medically or to sustain species.
Regarding your question about such prayer while wearing tefillin, among other aspects of tefillin is that it is part of the emphasis of translating ones thoughts and emotions into realized literal, physical, material action. Or in the words Rabbi Shimon ben Gamliel of Mishnah Avot 1:17, the action is the primary thing.
שִׁמְעוֹן בְּנוֹ אוֹמֵר, כָּל יָמַי גָּדַלְתִּי בֵין הַחֲכָמִים, וְלֹא מָצָאתִי לַגּוּף טוֹב אֶלָּא שְׁתִיקָה. וְלֹא הַמִּדְרָשׁ הוּא הָעִקָּר, אֶלָּא הַמַּעֲשֶׂה.
So no, if praying in the context described above, it would not be a problem to make such a prayer while wearing tefillin.