For a zygote to form, two haploid gametes undergo meiosis and fuse during fertilisation. Since two egg cells (or even two sperm cells) are both haploid, is it theoretically possible for them to make an embryo (via experimental manipulation)? I’m guessing that if such a thing were possible, the two egg cells could each come from the same organism or two different sources. Would there be any side effects, such as persistence of mutant alleles due to a lack of genetic variation if both egg cells originate from the same organism?
Yes, it is possible to make a zygote from two ovules. Other more complex scenarios are possible as well. You might want to read this cbc article. If the two ovules came from the same organism, then the baby will very likely suffer from a number of diseases due to very low heterozygosity (same mechanism that is causing inbreeding depression and problems of consanguinity).
It is likely not possible to get a zygote from two spermatozoids though because you would lack mitochondrial DNA (mtDNA). There is a little bit of mtDNA in spermatozoids but they are generally not transmitted. Through experimental setup I imagine it might be possible. Also, spermatozoids are very small and would lack many of the ressources necessary to start a zygote.
You're describing chimerism (distinct genotypes in a single organism; in animals, this arises when two fertilized eggs merge). This article suggests that "microchimerism" can occur during pregnancy when small quantities of fetal cells migrate through a mother's body. You can find additional examples from humans and other species at my first link (the Wikipedia article), but here's one more from The New England Journal of Medicine in early 1998:
A 3.46-kg infant was delivered vaginally at term; he had a normal right testis and an undescended left testis, with otherwise normal male genitalia. At the age of six months, the left testis was palpable at the inguinal ring. Surgical exploration at the age of 15 months revealed a hernial sac containing an abnormal gonad and vas deferens. These structures were excised; they proved on histologic examination to be an ovary with a fallopian tube attached to a horn of uterus. Karyotyping of peripheral-blood lymphocytes then revealed two cell lines, one 46,XX and the other 46,XY.