What is the usual manner for seeking costs orders for court judgments?
The usual manner is to ask for costs as part of the initiating pleading.
and can it be done after the fact... ?
As long as the court continues to have jurisdiction, usually until the court has issued its final judgment on the matter.
The rule at common law is that after the final order from the court on a matter, it is considered functus officio and without jurisdiction to do more.
Some provinces have provided a statutory/regulatory exception for costs orders after the final judgment on the substantive issues, where the final judgment was silent on costs. See e.g. Alberta Rules of Court, Alta Reg 124/2010, s. 10.30(1)(c) and its discussion in Kent v MacDonald, 2021 ABQB 953, at para. 27 and onward:
The decisions that rely upon Rule 10.30 as an exception to the functus principle when it comes to costs, deal with facts in which the court had not made any determination on costs in the initial decision. Rule 10.30 allows the Court to assess costs and issue an award after a decision is made; it does not allow for costs to be revisited repeatedly or varied.
Procedure will be based on the rules of the particular court, but normally, this would be by way of application or motion (to the judge who was seized of the case) through the ordinary manner of filing/serving/etc.