Is it okay to omit (one or multiple) middle names in a bibliography?

Bibliography entries should be faithful to the works to which they correspond: you should conform to the constraints of the style used by the journal, but you should not remove or add middle names or initials. The rationale for this is that doing otherwise makes it harder for readers to follow the citations back to the original literature if the names in the article are reported differently from the way they're indexed.

If the same author represented themselves differently in multiple works, that should be reflected in the bibliography entries, even if it makes things awkward for the author citing the works. If you feel you need to indicate that the authors are in fact the same individual, you may do so with a parenthetical comment.

It depends on the style manual you're following. I'll let someone else address APA.

CHICAGO MANUAL OF STYLE, 17TH EDITION

CMS Ch 14: Notes and Bibliography CMS 14.73: Form of author’s name

"Authors’ names are normally given as they appear with the source itself—that is, on the title page of a book or other stand-alone work or at the head of a journal article or the like. Certain adjustments, however, may be made to assist correct identification (but see 15.12). First names may be given in full in place of initials (but see 14.74). If an author uses his or her given name in one cited work and initials in another (e.g., “Mary L. Jones” versus “M. L. Jones”), the same form, preferably the fuller one, should be used in references to that author for both works. To help differentiate similar names, middle initials may be given where known. Degrees and affiliations following names on a title page are omitted."

CMS Ch 15: Author-Date References 15.12: Authors’ names in reference list entries

"In a reference list as in a bibliography, record the authors’ names as they appear on the title page or at the head of an article or chapter, with the exceptions noted in 14.72–84. Some publications, especially in the natural sciences, use initials rather than full given names (see 15.33). Where this practice is followed, an exception should be made where two authors share the same initials and last name."

MLA STYLE (Quoted from an old 2nd edition, confirmed by the current Perdue OWL) "Give the author's name as it appears on the title page. Never abbreviate a name given in full. . . . Use initials [only] if the title page does."

FOREIGN NAMES The Chicago Manual of Style has good introductory guidance on how to treat foreign names. The topic is complicated and sometimes you need to do some internet sleuthing to find more information.

For APA style you gnerally want the name to match how it is listed on the source:

Sometimes names are presented inconsistently across publications. If the author has used different forms of the same name on different works, then your reference list entries should match the form of the name on the work being cited for reasons of retrievability. For example, sometimes the author may use a middle initial and sometimes not (e.g., perhaps Jacob T. Baker sometimes publishes as Jacob Baker).

Because both names refer to the same person and the differences between names are minor (namely, a missing initial), it is not necessary to adjust the order of the works in the reference list to account for the missing initial or to put the author’s initials in the text citations to distinguish the references. (Read more about the order of works in the reference list and see examples.)

There are some nuances regarding the sort order in the bibliography and how to disambuate in text citations for APA style, and you should probably refer to your specific style guide.