It appears that the organisation does not have a global view of homosexuality, but each region may produce their own position statement.
I looked in detail my local region (Australian Territories), below. I also skimmed some of the other regions, see below.
The Salvation Army Australian Territories Positional Statement on Human Sexuality has some relevant sections:
They are against same-sex marriage:
The Salvation Army affirms its absolute conviction that the marriage of one man to one woman is a sacred institution ordained by God and that a traditional good-faith commitment to an indissoluble union is one of the most rewarding of life’s decisions for any man or woman, providing the optimal conditions for family life.
But against discriminating against people for not being in a nuclear family:
The Salvation Army acknowledges that many people today find themselves in situations outside the ideal stated above. Persons in such circumstances should always be encouraged to feel totally accepted within The Salvation Army.
They believe in celibacy outside of marriage:
We believe in chastity before marriage and fidelity within the marriage relationship.
Note: The position that marriage is between a man and a woman, and that sex should be restricted to married couples has implications for same-sex couples.
They oppose homosexual practice:
It is The Salvation Army’s belief that whilst recognising the possibility of such orientation, (the origins of which are uncertain), the Bible expressly opposes homosexual practice, seeing such activity as rebellion against God’s plan for the created order.
They oppose vilification or discrimination against non-practising homosexuals:
A disposition towards homosexuality is not in itself blameworthy nor is the disposition seen as rectifiable at will. The Army is sensitive to the complex social, emotional and spiritual needs of all people including those with homosexual inclinations. We oppose vilification of, or discrimination against anyone on the grounds of sexual orientation. No person is excluded from membership, fellowship or service in the Army solely on the basis of sexual disposition.
However, practicing homosexuals are not permitted to become full members:
Homosexual practice, however, is, in the light of Scripture, clearly unacceptable. Such activity is chosen behaviour and is thus a matter of the will. It is therefore able to be directed or restrained in the same way heterosexual urges are controlled. Homosexual practice would render any person ineligible for full membership (soldiership) in the Army. However, practising homosexuals are welcome to worship with, and join in the fellowship of The Salvation Army.
Summary: The official position of the Australian Salvos is clearly not completely accepting of active gay and lesbian people. However, it is tempered with some attempts at tolerance, if not acceptance.
Other regions have similar positions, but must be read individually. A sample include:
There are significant differences in the wording. For example, the Indian version has no explicit statement that practising homosexuals are welcomed to worship with the Salvation Army. The Canada/Bermuda statement is even less explicit about their policies. I was unable to find the UK version. I did not attempt to find anything for non-English speaking countries, sorry.
This answer does not attempt to address any differences between the official policies of the organisation, the actual behaviour of their agents while representing the organisation or the behaviour of those same individuals when outside their role.