The law certainly may not constitutionally overturn Obergefell v. Hodges, the U.S. Constitution, or any applicable federal law, or the state constitution.
And, while a particularly individual county clerk could not be fired for refusing to issue a same-sex couple a marriage licenses, it is not at all obvious that a county clerk's office itself, could refuse to allow anyone in the office to issue that marriage license.
On the other hand, many forms of discrimination by private parties, e.g. in employment, educational admissions, public accommodations, etc. are not directly prohibited by federal law.
Some federal grant programs, however, that revenue share with state and local governments, either do, or might in the future have, a requirement to respect LGBT rights with respect to funded programs. Generally, a state can refuse to participate in those federal programs in order to avoid those strings, but Wyoming would be preventing its state and local government officials from obtaining funds that would be available in other states that were willing to comply with those strings.
It does effectively prohibit local non-discrimination ordinances to have much force because they would have to have a broad religious conviction exception (although people who didn't express a religion based reason for discrimination could still be bound by local ordinances).
This is probably fatal to the legislation under Romer v. Evans, 517 U.S. 620 (1996) in which the U.S. Supreme Court held that a law preventing local governments from enacting ordinances protecting LGBT rights as this was a matter of exclusively state concern was unconstitutional as it denied one class of persons the right to seek protection from local government, while affording it to others.
But, HB-135 attempts to overcome Romer v. Evans, by requiring anyone who does discriminate against LGBT people to have a basis in religious conviction to do so, effectively attempting to pit the 14th Amendment against the 1st Amendment's free exercise clause.
On the other hand, since Romer v. Evans focused largely on the impairment of the political rights of people who want to protect gay rights (regardless of their own sexual orientation) to seek protection on the same basis as they could seek other laws, the limitation that discrimination must be religiously justified to be protected might not be sufficient to distinguish the law.
In some circumstances, such as denying a local government the ability to prevent a church that discriminates for religious reasons from renting athletic fields owned by the government, a free exercise argument might prevail.
In others, such as using a free exercise argument to prevent anyone in a county clerk's office from issuing a marriage license to a same sex couple, it would almost certainly fail.
HB-135 is deliberately crafted to address circumstances that haven't been precisely tested by prior precedents to push the envelope and to bring test cases to the U.S. Supreme Court. But, given the very broad pro-gay rights opinion language in the 10th Circuit case mandating that same-sex marriage be permitted prior to Obergefell, and in Obergefell itself in the U.S. Supreme Court, it is very likely that HB-135 would at a minimum not be upheld in all of the circumstances that it purports to apply to, although it might also not be held facially unconstitutional in any and all circumstances.
For example, free exercise protections might be very high for formally religious organizations like churches and seminaries, might be intermediate when a closely held entity's owner and officers have particular religious beliefs (in light of the Hobby Lobby case related to reproductive health care mandates of the ACA) when the company serves the general public or an educational institution that teaches secular subjects has a religious affiliation, and might be very weak in the case of publicly held companies and regulated utilities.
The effective impact of the law on particular LGBT individuals as applied would also be relevant.
For example, if the only homeless shelters in Casper were operating by religiously affiliated organizations that received government grants, and this law had the practical effect of denying any shelter to any transgender homeless person in Casper, there is a good chance that free exercise would not prevail over equal protection. On the other hand, If Casper had both a secular Volunteers of America shelter and a religious Salvation Army shelter, both funded by government grants, and the effect of the law with to limit the shelter choices of a transgender homeless person in Casper, rather than to deny that person access to any shelter, free exercise might prevail over equal protection.
So, the law might very well be found unconstitutional in some circumstances but not others, as applied, even if it were not found to be facially invalid under an argument similar to the one made in Romer.