This is an excellent question. I don't have a great answer and I hope it will get more posts.
I am acquainted with some ideas on this topic; some, but not all, through chassidus Chabad. Here are a few of them:
1. Get the mitzvah item at any cost.
This is what the Jews of old did. I include with them the Jews of the last century. The stories of their courage and mesiras nefesh in obtaining and preserving the needed articles, especially during the Holocaust, are extremely moving. Of course, this was always true, not only in Europe but for thousands of years before. This is no longer remarkable to the Jewish ear because we have heard it so much because it was done so much.
2. "If you can't, you can't"
There is such an idea as not being able to perform a mitzvah, and it very likely stops short of that, where other mitzvot such as pikuach nefesh or the Decree of Usha or similar checks from the Torah come in to play before you actually go to the end of human effort and find that you "can't." One is allowed to stop pursuing a mitzvah in these cases, and sometimes it is required.
Seen another way: We try as hard as we can, and count on H' for the rest. That is the proper way of doing everything. In the latest parshah (Pikudei) we saw H' command Moshe to lift the beams of the mishkan, which were physically impossible to lift; Moshe lifted them as high as he could, and H' lifted them the rest of the way. In other words we have faith that Siyata Dishmaya will narrow the gap between our abilities and what He has asked of us, and that G-d's mercy will close it. ("If you can't, you can.")
Second chances, and opportunities for repair and teshuvah, exist in Judaism. In some cases they are enshrined in the tradition (tashlumin, Pesach Sheni, vidui and tachanun, korbanos, Yom Kippur, etc., etc.), and, where they are not, we take the standard steps of teshuvah (apologies, repairing damage, contrition, commitment to do better) and then look forward with clear eyes.
There are specific tikkunim such as personal fasts and tehillim that supposedly repair the spiritual damage of a mistake. The Lubavitcher Rebbe seemed to think, more practically, that atonement lay in making the next time better. He would suggest to chasidim who asked him for a tikkun that they spend time learning Torah about the area they had transgressed.
4. Doing another form of the mitzvah
There is some idea in our tradition of doing physical versions of mitzvot similar to those you cannot--as in using zeroa to stand in for the Peasch, and a personal fast to stand in for a korban. At least as interestingly, there are examples brought in our tradition of mitzvot that may be performed in spiritual form--either by learning, by davening, or by having a machshava--as an adjunct to the physical mitzvah or even as its replacement. The example repeated most throughout Jewish tradition is bringing korbanos, which the Talmud says in several places may be replaced by studying their halachos [Taanis 27b, Menachos 110a, Yoma 86b, and Vayikra Rabbah 7:3, which says: "Since you involve yourself in studying them, I consider it as though you had actually brought them."] According to various sources, the mitzvah of building the Temple is achieved by learning the halachos of building the Temple and/or by making a place for Hashem in our homes, hearts, and synagogues.
The maamar Isa B'Midrash Tehillim of the Fifth Rebbe of Chabad explores how a spiritually- or intellectually-attained mitzvah may accomplish the same spiritual tasks as a physical mitzvah. This idea of the "long-short way" in Chasidic practice, where a mitzvah is ideally accomplished through feeling, understanding, and self-transformation above and beneath the doing, is rehearsed in Tanya and throughout Chasidus Chabad. The existence of purely-spiritual forms of the mitsvot (all of them!) at the place of their initiation, namely the Supernal Realms, is the source of this idea. (I think I also learned that in the cases where the avos couldn't do a particular mitzvah because it or its conditions didn't yet exist, they did the spiritual form of the mitzvah, too.) It therefore seems that performing the spiritual equivalent of a mitzvah--namely, learning about it--is a legitimate option when one has no way to do the deed itself.
5. Paying someone else to do the mitzvah
I heard this was also recommended by the Lubavitcher Rebbe in a case when a chosid had missed an opportunity for a mitzvah. He suggested the chosid give some money afterwards to an organization working to further this mitzvah. This seems like a very practical offset to me.
Also, many mitzvot--like matones l'evyonim, writing a Torah scroll, kibbud av va'eim, and perhaps even certain prayers--may be performed through a shaluach or a motzi. Let us hope that the shaluach or the motzi has the ritual object that you lack.
6. Doing a different mitzvah
“A mitzvah does not atone for a sin” (Sforno, Exod. 32:33)--and yet a mitzvah is the right thing to be doing at all times. Perhaps, instead of the impossible mitzvah, one could fulfill another one, like one that one had neglected. There is maybe even a tiny bit of backup for this advice in the Torah:
All mitzvot nourish the entire soul, and there are mitzvot that are obligated for all times. If you can't do the mitzvah you are supposed to be doing, at least you will be doing another mitzvah.
7. The effort to do the mitzvah is the mitzvah
To recapitulate number 2, what you're supposed to do is try. The time you spend trying is avodas hakodesh and whatever happens in the end is, strictly speaking, not your problem.
Everything I have just said and infinitely more is summarized most gracefully in this six-sentence letter of the Lubavitcher Rebbe to one of his chasidim.
And yet one more thing...
I find myself smiling at the apt and lovely fact that G-d anticipated your exact question to the letter and supplied a plain answer in His torah. (Isn't that always the case on Mi.Yodeya, as everywhere?!)
That answer, of course, is the story of the Chanukah miracle, in which people desperately wanted to do a mitzvah, but found themselves lacking the goods for it. What happened was that G-d provided the goods, much as He provided the lamb, and provides everything. The Torah was given to people; people live on Earth; Earth is defined by limitations. We, unlike angels, sometimes don't have the goods. And yet G-d makes a way for us to serve Him, as if to prove that He He wants us to serve Him indeed. We continue to celebrate that gift for the rest of our generations, as if to prove, as if it were necessary, that we really want to serve Him, too.