TL;DR: look up Mahajan's Generalization regarding the effect alignment has on constituent order.
Survey of Alignment
It's important to recognize that there are several ways in which a language can be ergative, for example:
- syntactically: "father returned and _ saw mother" implies in nominative alignment (e.g. English) that "father saw mother", but in ergative languages it implies "mother saw father".
- in derivational morphology, compounding etc.: e.g. in English "baby-sit" the baby is being baby-sitted and not doing the baby-sitting, i.e. the object gets incorporated.
- pronouns: in nominative alignment like English we have "he runs" and "I see him", i.e. the intransitive subject is marked like the transitive agent; in ergative languages one would instead have "him runs" and "I see him", i.e. the intransitive subject is marked like the transitive patient.
- nominal morphology: similar to the pronominal case in that nominative languages have "boy-NOM run" and "boy-NOM see car-ACC" whereas ergative languages have "boy-ABS run" and "boy-ERG see car-ABS"; however, many ergative languages use ergative morphology only on nouns and have nominative alignment on the pronouns since pronouns tend to be higher in the animacy hierarchy, i.e. are more likely to be agents.
- verbal marking: a nominative language might have "boy sees-3SG me" whereas an ergative might have "boy sees-1SG me" (or, as in the example on the link, if the intransitive subject is a prefix to the verb then a nominative alignment would prefix the agent and suffix the patient whereas ergative alignment would be the reverse).
Survey of Constituent Order
Now, WALS apparently offers us no information on syntactic alignment; but on the last three points it offers information we can use to determine correlation between constituent order and alignment. With respect to constituent order we know that
- the order of subject, object and verb is most commonly SOV or SVO, rarely VSO, even less VOS and OVS and most rarely OSV.
- the order of subject and object tends to be SV, only rarely VS.
- the order of object and verb can go either way.
Putting it Together
Now we combine these features of constituent order with alignment to determine whether there is anything interesting to see:
- nominal morphology:
- order of subject and verb: where for nominative alignment SV is preferred by a factor of about 20, it's only a factor of 5 for ergative languages; however, the low number of languages in the language makes this not very reliable.
- order of object and verb: languages with non-nominative alignment tend more toward OV than languages with nominative languages.
- order of subject, object and verb: SOV appears to be strongly favored (accounting for about 80% of languages for each alignment) by both nominative and accusative alignment - about 80% of SVO occurences come from languages with neutral alignment. This is because for SVO order the position relative to the verb determines the role of the argument, whereas for SOV languages the role gets marked.
- pronouns:
- order of subject and verb: independently of alignment languages seem to prefer SV by a factor of about 5.
- order of object and verb: languages with non-nominative alignment tend more toward OV than languages with nominative languages.
- order of subject, object and verb: where nominative alignment shows preference first for SOV, then SVO and finally VSO, ergative alignment does also prefer SOV to VSO, but does not show SVO order. Also note that neutral alignment, unlike nominative alignment, prefers SVO to SOV.
- verbal marking:
- order of subject and verb: where for nominative alignment SV is preferred by a factor of about 10, it's only a factor of 2 for ergative languages; however, the low number of languages in the language makes this not very reliable.
- order of object and verb: languages with non-nominative alignment tend slightly more toward OV than languages with nominative languages.
- order of subject, object and verb: where nominative alignment shows preference first for SOV, then SVO and finally VSO, ergative alignment does also prefer SOV to VSO, but does not show SVO order.
Summary
- order of object and verb: independently of alignment a strong tendency toward SV can be observed; however, for nominative alignments the strength of the tendency is about twice as strong as for non-nominative alignments which are more likely to permit VS.
- order of subject and object: for languages with nominative alignment both VO and OV are observed in approximately equal number (with a slight preference toward VO), whereas languages with non-nominative alignment seem to strongly prefer OV - however, whereas that preference is strong, the number of non-nominative languages in the sample is too small to make this statistically reliable.
- order of subject, object and verb: where languages with nominative alignment prefer constituent orders SOV, SVO and VSO (in that order), ergative alignment does not with SVO order (at least in fully ergative languages); this is known as Majahan's Generalization. Additionally it seems as if ergative languages have a greater tendency to have no dominant word order but more variable word order.
Note that those observation might well be statistical artifacts - I'll leave further research to you.
Further Reading
If you want to know about alternate word orders permitted within one language (e.g. you can say "him I saw", focusing the object to call attention to it) you'll presumably have to look at individual languages.
According the data from WALS, the following languages might be relevant: