Many languages I know of have a way of making causative constructions. For example, English uses "make" or "have": I make you do something or have you do something, or even cause you to do something. How do you do that in Latin?
A relatively common construction in Latin is to use the verb curare. If "I do this" is hoc facio, then "I have someone do this" is "hoc faciendum curo". To my knowledge this construction is impersonal in the sense that you cannot indicate who you make do the job.
There is a way to indicate who has to do something, and it is called the passive periphrastic conjugation and formed with esse, dative and gerundive. For example, hoc tibi faciendum est is "you have to do this". This structure does not typically indicate why the thing has to be done, but a causal ablative can always be added:
Meo iusso hoc tibi faciendum est.
"Due to my order you have to do this."
This is not far from saying te hoc facere iubeo or "I order you to do this". Therefore it does not feel like the causative as much as you might want.
A heavier option is to use a final ut clause with efficere or some other suitable verb:
Effeci, ut hoc faceres. — Effeci, ut hoc tibi faciendum esset.
I did so that you do this. — I did so that you have to do this.
The English translations are a bit clumsy, but I wanted to emphasize the Latin structure.
I think none of these ideas exactly matches either of the English causative structures you mention, but they should offer a decent toolbox for expressing similar ideas in Latin.
In medieval Latin, habere came to express (also) ability or obligation. I'd have to dig up an example, but I see it in Ockham with some (minor) frequency.
However, it is also mentioned in Elliott's 'Brief Introduction to Medieval Latin Grammar' in Medieval Latin, ed. Harrington, rev. Pucci (Chicago, 1997), § 7.2.2, citing (e.g.) Tertullian, Apologeticum 37:
Si inimicos, ut supra diximus, iubemur diligere, quem habemus odisse?
Edit. As pointed out in a comment, the above construction is not, strictly speaking, a 'causative clause'.
As far as I know, there is no special 'clause' in Latin that qualifies, but consider this examples (from Bradley's Arnold, § 131):
Per te stetit quominus vinceremus (You were the cause of our not winning the day)
In the 'General Vocabulary', he also gives:
per me fit ut = I am the cause of
More common, I think, would be the following (based on examples in Cassell's Latin Dictionary):
facere or efficere plus ut and the subjunctive;
cogere plus the infinitive;
movere -- risum homini movere (to make a person laugh);
dare and exhibere (usually translated, in this context, as producing/causing something -- not making someone do something)
The two examples in Joonas Ilmavirta's answer are also quite relevant, of course.
In addition to the excellent answers, I'll add the construction facere ut + subjunctive.
We see it all over in Plautus, e.g. Bacchides:
Propterea hoc facio, ut suadeas gnato meo
ut pergraecetur tecum, tervenefice.
All over in Cicero, e.g. Pro Cælio:
Quæ tu omnia tuis foedis factis facis ut nequiquam velim: vix ferendi.
And so on. The construction seems, as far as a quick look through the corpus indicates, to be used more impersonally than personally, but personal uses abound.
I'm unsure to what extent this was used, but one can find the following causative construction, facere + inf., in Book II of the Aeneid:
Quī nātī cōram mē cernere lētum / fēcistī [...]
You, who made me watch the death of my son
I would perhaps be cautious about using this outside the realm of poetry; Lewis & Short, also citing Vergil, lists this construction as "rare" (under I.A.γ):
With inf. = efficere, curare, to cause (rare): nulla res magis talis oratores videri facit, Cic. Brut. 38, 142; Pall. 6, 12: aspectus arborum macrescere facit volucres inclusas, Varr. R. R. 3, 5, 3; Sall. Fragm. ap. Sen. Ep. 114: qui nati coram me cernere letum Fecisti, Verg. A. 2, 539; Ov. H. 17, 174: mel ter infervere facito, Col. 12, 38, 5 (perh. also in Ov. H. 6, 100, instead of favet, v. Loers. ad h. l.; cf. infra, B. 4.).