The Talmud (RH 4a) tells us that Tzedaka, like any other Neder, is included in the Biblical prohibition of Bal Te'acher (don't delay paying up your vows). However, unlike ordinary vows for sacrifices for which one does not violate unless they don't bring the sacrifice before 3 consecutive festivals have passed (Rambam Maaseh Korbanot 14:13), the Talmud tells us (RH 6a) that for Tzedaka one needs to pay immediately for there are always poor people available. Rambam codifies this in Matnot Aniyim 8:1, noting that if there is no poor person available, one seperates the money and awaits a poor person. Tosfot (RH 4a) explains that while the 3-festival rule doesn't apply when there are needy people available, if there are no needy people then you have to seek them out before 3 festivals have elapsed. Rambam (ibid.) mentions no such distinction. The Tur (YD 257) quotes his father, Rosh, as arguing on the Rambam and ruling that if one were to stipulate at the time of the Neder that one should be allowed to distribute the money slowly to different people, that would avoid any of these limits.
Shulchan Arukh YD 257:3 rules like the Rambam against Tosfot (see Shakh 5, Gra 6), and like the Rosh against the Rambam. However, in your case we need not worry about any of that, as the Rama there notes (my translation; bolding added):
הגה: וכל זה בצדקה שיש בידו לחלקה בעצמו אבל כשנודרין צדקה בבית הכנסת ליתנה ליד גבאי או שאר צדקה שיש לו ליתן לגבאי אינו עובר עליה אע"ג דעניים מצויים אא"כ תבעו הגבאי ואז עובר עליה מיד אי קיימי עניים והגבאי היה מחלק להם מיד ואם אין ידוע לגבאי צריך הוא להודיע לגבאי מה שנדר כדי שיוכל לתבעו ... ואם אמר אתן סלע לצדקה לפלוני אינו עובר עד שיבא אותו עני אע"ג דשאר עניים מצויים:
GLOSS: All this is by Tzedaka which is in his capability to distribute, but when they vow Tzedaka in synagogue to give to the hands of the Gabbai or other Tzedaka which one needs to give to the Gabbai [for distribution], he does not violate even if poor people are present unless the Gabbai sought [the money from] him, and then he violates it immediately if the poor people are present and the Gabbai would distribute to them immediately. If [his vow] was not known to the Gabbai, he must notify him so that he can seek [the money from] him. ... And if he says "I will give a Sela to [a specific poor person" he does not violate until that specific person comes, even if there are other poor people present.
It seems from this that not only are the people who put money in the box not bound by time limits, but the box manager too is not bound by any limit as it is not up to him to distribute the money. If the organization requests the money (the Shakh there states that until otherwise known we assume they need it for immediate use) then he must (a biblical obligation!) send it to them immediately. If they haven't requested the money (and he doesn't want to send it yet), he should notify them that the money is available so they can request it if they want. Seemingly, he can assume they don't want to be bothered with notifications for each coin dropped in.